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### Dates
DFS discs do not track any dates (because Acorn MOS prior to version 3 did not maintain a real-time clock) but instead offer a peculiar feature: a modification count. Every time the catalogue is updated, the count increments. The count is shown in parentheses after the title in the first line of the disc catalogue, such as the `12` in the catalogue listing shown earlier.
Other features
The DFS also supports a means to start up disc software based on a key sequence. If the shift key is held while the machine is soft or hard reset, the DFS checks drive 0 for a disc containing a positive boot flag. The boot flag is either 0 (ignore), 1 (load file), 2 (run machine code file) or 3 ("execute" script). If the boot flag is positive, a file called `$.!BOOT` is looked for and loaded into memory (1), loaded and executed as machine code (2) or fed into the keyboard buffer (3). Option 3 reads "EXEC" files, text macro files used as primitive shell scripts. These are not true shell scripts but simply a series of keys to be typed, like a recording to play back. Thus, they cannot loop or branch unless they input such code into the BASIC interpreter. As well as being used during a reset, they can be executed at any time with the operating system's `*EXEC` command.
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Thus, they cannot loop or branch unless they input such code into the BASIC interpreter. As well as being used during a reset, they can be executed at any time with the operating system's `*EXEC` command. EXEC files are file system independent.
## Alternatives
There was a variant of the DFS called the DNFS, or Disc/Network Filing System, that contained the Econet Network Filing System (NFS), standard Disc Filing System and Tube co-processor support software on a single ROM; this ROM installed two filing systems into the OS at once.
The initial design for the DFS was based around an Intel Corporation FDC 8271 disc drive controller, the immediate predecessor of the 8272 design found in the IBM Personal Computer. The 8271 controller was of limited functionality and obsolete, and later versions of the file system from various vendors including Solidisk, Acorn and Watford Electronics were based on the later WD1770 and WD1772 drive controllers. The 1770 controller quickly became the standard controller in the Acorn range following the advent of ADFS.
The DFS was superseded by the Advanced Disc Filing System (ADFS) which was fully hierarchical and was suitable for running hard drives on the BBC Micro. The ADFS was the default filing system on the BBC Master but most users are likely to have reverted to the DFS for compatibility reasons; ADFS could not read DFS discs.
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The DFS was superseded by the Advanced Disc Filing System (ADFS) which was fully hierarchical and was suitable for running hard drives on the BBC Micro. The ADFS was the default filing system on the BBC Master but most users are likely to have reverted to the DFS for compatibility reasons; ADFS could not read DFS discs. The ADFS was also chosen as the standard filing system for RISC OS.
## References
## External links
- Acorn, Watford and Duggan DFS file system structure
- Watford DFS (extended Acorn DFS) specification - includes the original Acorn DFS specification
- Acorn DFS manual (RTF format)
Category:Disk file systems
Category:Acorn Computers
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In data analysis, anomaly detection (also referred to as outlier detection and sometimes as novelty detection) is generally understood to be the identification of rare items, events or observations which deviate significantly from the majority of the data and do not conform to a well defined notion of normal behavior. Such examples may arouse suspicions of being generated by a different mechanism, or appear inconsistent with the remainder of that set of data.
Anomaly detection finds application in many domains including cybersecurity, medicine, machine vision, statistics, neuroscience, law enforcement and financial fraud to name only a few. Anomalies were initially searched for clear rejection or omission from the data to aid statistical analysis, for example to compute the mean or standard deviation. They were also removed to better predictions from models such as linear regression, and more recently their removal aids the performance of machine learning algorithms. However, in many applications anomalies themselves are of interest and are the observations most desirous in the entire data set, which need to be identified and separated from noise or irrelevant outliers.
Three broad categories of anomaly detection techniques exist. Supervised anomaly detection techniques require a data set that has been labeled as "normal" and "abnormal" and involves training a classifier.
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Three broad categories of anomaly detection techniques exist. Supervised anomaly detection techniques require a data set that has been labeled as "normal" and "abnormal" and involves training a classifier. However, this approach is rarely used in anomaly detection due to the general unavailability of labelled data and the inherent unbalanced nature of the classes. Semi-supervised anomaly detection techniques assume that some portion of the data is labelled. This may be any combination of the normal or anomalous data, but more often than not, the techniques construct a model representing normal behavior from a given normal training data set, and then test the likelihood of a test instance to be generated by the model. Unsupervised anomaly detection techniques assume the data is unlabelled and are by far the most commonly used due to their wider and relevant application.
## Definition
Many attempts have been made in the statistical and computer science communities to define an anomaly. The most prevalent ones include the following, and can be categorised into three groups: those that are ambiguous, those that are specific to a method with pre-defined thresholds usually chosen empirically, and those that are formally defined:
### Ill defined
- An outlier is an observation which deviates so much from the other observations as to arouse suspicions that it was generated by a different mechanism.
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The most prevalent ones include the following, and can be categorised into three groups: those that are ambiguous, those that are specific to a method with pre-defined thresholds usually chosen empirically, and those that are formally defined:
### Ill defined
- An outlier is an observation which deviates so much from the other observations as to arouse suspicions that it was generated by a different mechanism.
- Anomalies are instances or collections of data that occur very rarely in the data set and whose features differ significantly from most of the data.
- An outlier is an observation (or subset of observations) which appears to be inconsistent with the remainder of that set of data.
- An anomaly is a point or collection of points that is relatively distant from other points in multi-dimensional space of features.
- Anomalies are patterns in data that do not conform to a well-defined notion of normal behaviour.
### Specific
- Let T be observations from a univariate Gaussian distribution and O a point from T. Then the z-score for O is greater than a pre-selected threshold if and only if O is an outlier.
## History
###
### Intrusion detection
The concept of intrusion detection, a critical component of anomaly detection, has evolved significantly over time.
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## History
###
### Intrusion detection
The concept of intrusion detection, a critical component of anomaly detection, has evolved significantly over time. Initially, it was a manual process where system administrators would monitor for unusual activities, such as a vacationing user's account being accessed or unexpected printer activity. This approach was not scalable and was soon superseded by the analysis of audit logs and system logs for signs of malicious behavior.
By the late 1970s and early 1980s, the analysis of these logs was primarily used retrospectively to investigate incidents, as the volume of data made it impractical for real-time monitoring. The affordability of digital storage eventually led to audit logs being analyzed online, with specialized programs being developed to sift through the data. These programs, however, were typically run during off-peak hours due to their computational intensity.
The 1990s brought the advent of real-time intrusion detection systems capable of analyzing audit data as it was generated, allowing for immediate detection of and response to attacks. This marked a significant shift towards proactive intrusion detection.
As the field has continued to develop, the focus has shifted to creating solutions that can be efficiently implemented across large and complex network environments, adapting to the ever-growing variety of security threats and the dynamic nature of modern computing infrastructures.
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This marked a significant shift towards proactive intrusion detection.
As the field has continued to develop, the focus has shifted to creating solutions that can be efficiently implemented across large and complex network environments, adapting to the ever-growing variety of security threats and the dynamic nature of modern computing infrastructures.
## Applications
Anomaly detection is applicable in a very large number and variety of domains, and is an important subarea of unsupervised machine learning. As such it has applications in cyber-security, intrusion detection, fraud detection, fault detection, system health monitoring, event detection in sensor networks, detecting ecosystem disturbances, defect detection in images using machine vision, medical diagnosis and law enforcement.
Intrusion detection
Anomaly detection was proposed for intrusion detection systems (IDS) by Dorothy Denning in 1986. Anomaly detection for IDS is normally accomplished with thresholds and statistics, but can also be done with soft computing, and inductive learning. Types of features proposed by 1999 included profiles of users, workstations, networks, remote hosts, groups of users, and programs based on frequencies, means, variances, covariances, and standard deviations. The counterpart of anomaly detection in intrusion detection is misuse detection.
### Fintech fraud detection
Anomaly detection is vital in fintech for fraud prevention.
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The counterpart of anomaly detection in intrusion detection is misuse detection.
### Fintech fraud detection
Anomaly detection is vital in fintech for fraud prevention.
### Preprocessing
Preprocessing data to remove anomalies can be an important step in data analysis, and is done for a number of reasons. Statistics such as the mean and standard deviation are more accurate after the removal of anomalies, and the visualisation of data can also be improved. In supervised learning, removing the anomalous data from the dataset often results in a statistically significant increase in accuracy.
### Video surveillance
Anomaly detection has become increasingly vital in video surveillance to enhance security and safety. With the advent of deep learning technologies, methods using Convolutional Neural Networks (CNNs) and Simple Recurrent Units (SRUs) have shown significant promise in identifying unusual activities or behaviors in video data. These models can process and analyze extensive video feeds in real-time, recognizing patterns that deviate from the norm, which may indicate potential security threats or safety violations. An important aspect for video surveillance is the development of scalable real-time frameworks. Such pipelines are required for processing multiple video streams with low computational resources.
### IT infrastructure
In IT infrastructure management, anomaly detection is crucial for ensuring the smooth operation and reliability of services.
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Such pipelines are required for processing multiple video streams with low computational resources.
### IT infrastructure
In IT infrastructure management, anomaly detection is crucial for ensuring the smooth operation and reliability of services. These are complex systems, composed of many interactive elements and large data quantities, requiring methods to process and reduce this data into a human and machine interpretable format . Techniques like the IT Infrastructure Library (ITIL) and monitoring frameworks are employed to track and manage system performance and user experience. Detected anomalies can help identify and pre-empt potential performance degradations or system failures, thus maintaining productivity and business process effectiveness.
### IoT systems
Anomaly detection is critical for the security and efficiency of Internet of Things (IoT) systems. It helps in identifying system failures and security breaches in complex networks of IoT devices. The methods must manage real-time data, diverse device types, and scale effectively. Garbe et al. have introduced a multi-stage anomaly detection framework that improves upon traditional methods by incorporating spatial clustering, density-based clustering, and locality-sensitive hashing. This tailored approach is designed to better handle the vast and varied nature of IoT data, thereby enhancing security and operational reliability in smart infrastructure and industrial IoT systems.
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Garbe et al. have introduced a multi-stage anomaly detection framework that improves upon traditional methods by incorporating spatial clustering, density-based clustering, and locality-sensitive hashing. This tailored approach is designed to better handle the vast and varied nature of IoT data, thereby enhancing security and operational reliability in smart infrastructure and industrial IoT systems.
### Petroleum industry
Anomaly detection is crucial in the petroleum industry for monitoring critical machinery. Martí et al. used a novel segmentation algorithm to analyze sensor data for real-time anomaly detection. This approach helps promptly identify and address any irregularities in sensor readings, ensuring the reliability and safety of petroleum operations.
### Oil and gas pipeline monitoring
In the oil and gas sector, anomaly detection is not just crucial for maintenance and safety, but also for environmental protection. Aljameel et al. propose an advanced machine learning-based model for detecting minor leaks in oil and gas pipelines, a task traditional methods may miss.
## Methods
Many anomaly detection techniques have been proposed in literature. The performance of methods usually depend on the data sets. For example, some may be suited to detecting local outliers, while others global, and methods have little systematic advantages over another when compared across many data sets.
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The performance of methods usually depend on the data sets. For example, some may be suited to detecting local outliers, while others global, and methods have little systematic advantages over another when compared across many data sets. Anomaly detection benchmark data repository of the Ludwig-Maximilians-Universität München; Mirror at University of São Paulo. Almost all algorithms also require the setting of non-intuitive parameters critical for performance, and usually unknown before application. Some of the popular techniques are mentioned below and are broken down into categories:
### Statistical
#### Parameter-free
Also referred to as frequency-based or counting-based, the simplest non-parametric anomaly detection method is to build a histogram with the training data or a set of known normal instances, and if a test point does not fall in any of the histogram bins mark it as anomalous, or assign an anomaly score to test data based on the height of the bin it falls in. The size of bins are key to the effectiveness of this technique but must be determined by the implementer.
A more sophisticated technique uses kernel functions to approximate the distribution of the normal data. Instances in low probability areas of the distribution are then considered anomalies.
#### Parametric-based
- Z-score,
- Tukey's range test
- Grubbs's test
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Instances in low probability areas of the distribution are then considered anomalies.
#### Parametric-based
- Z-score,
- Tukey's range test
- Grubbs's test
### Density
- Density-based techniques (k-nearest neighbor, local outlier factor, isolation forests, and many more variations of this concept)
- Subspace-base (SOD), correlation-based (COP) and tensor-based outlier detection for high-dimensional data
- One-class support vector machines (OCSVM, SVDD)
### Neural networks
- Replicator neural networks, autoencoders, variational autoencoders, long short-term memory neural networks
- Bayesian networks
- Hidden Markov models (HMMs)
- Minimum Covariance Determinant
- Deep Learning
- Convolutional Neural Networks (CNNs): CNNs have shown exceptional performance in the unsupervised learning domain for anomaly detection, especially in image and video data analysis. Their ability to automatically and hierarchically learn spatial hierarchies of features from low to high-level patterns makes them particularly suited for detecting visual anomalies. For instance, CNNs can be trained on image datasets to identify atypical patterns indicative of defects or out-of-norm conditions in industrial quality control scenarios.
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Their ability to automatically and hierarchically learn spatial hierarchies of features from low to high-level patterns makes them particularly suited for detecting visual anomalies. For instance, CNNs can be trained on image datasets to identify atypical patterns indicative of defects or out-of-norm conditions in industrial quality control scenarios.
- Simple Recurrent Units (SRUs): In time-series data, SRUs, a type of recurrent neural network, have been effectively used for anomaly detection by capturing temporal dependencies and sequence anomalies. Unlike traditional RNNs, SRUs are designed to be faster and more parallelizable, offering a better fit for real-time anomaly detection in complex systems such as dynamic financial markets or predictive maintenance in machinery, where identifying temporal irregularities promptly is crucial.
- Foundation models: Since the advent of large-scale foundation models that have been used successfully on most downstream tasks, they have also been adapted for use in anomaly detection and segmentation. Methods utilizing pretrained foundation models inclue using the alignment of image and text embeddings (CLIP, etc.) for anomaly localization, while others may use the inpainting ability of generative image models for reconstruction-error based anomaly detection.
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- Foundation models: Since the advent of large-scale foundation models that have been used successfully on most downstream tasks, they have also been adapted for use in anomaly detection and segmentation. Methods utilizing pretrained foundation models inclue using the alignment of image and text embeddings (CLIP, etc.) for anomaly localization, while others may use the inpainting ability of generative image models for reconstruction-error based anomaly detection.
### Cluster-based
- Clustering: Cluster analysis-based outlier detection
- Deviations from association rules and frequent itemsets
- Fuzzy logic-based outlier detection
### Ensembles
- Ensemble techniques, using feature bagging, score normalization and different sources of diversity
### Others
Histogram-based Outlier Score (HBOS) uses value histograms and assumes feature independence for fast predictions.
## Anomaly detection in dynamic networks
Dynamic networks, such as those representing financial systems, social media interactions, and transportation infrastructure, are subject to constant change, making anomaly detection within them a complex task. Unlike static graphs, dynamic networks reflect evolving relationships and states, requiring adaptive techniques for anomaly detection.
### Types of anomalies in dynamic networks
1. Community anomalies
1. Compression anomalies
1. Decomposition anomalies
1. Distance anomalies
1.
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Decomposition anomalies
1. Distance anomalies
1. Probabilistic model anomalies
## Explainable anomaly detection
Many of the methods discussed above only yield an anomaly score prediction, which often can be explained to users as the point being in a region of low data density (or relatively low density compared to the neighbor's densities). In explainable artificial intelligence, the users demand methods with higher explainability. Some methods allow for more detailed explanations:
- The Subspace Outlier Degree (SOD) identifies attributes where a sample is normal, and attributes in which the sample deviates from the expected.
- Correlation Outlier Probabilities (COP) compute an error vector of how a sample point deviates from an expected location, which can be interpreted as a counterfactual explanation: the sample would be normal if it were moved to that location.
## Software
- ELKI is an open-source Java data mining toolkit that contains several anomaly detection algorithms, as well as index acceleration for them.
- PyOD is an open-source Python library developed specifically for anomaly detection.
- scikit-learn is an open-source Python library that contains some algorithms for unsupervised anomaly detection.
- Wolfram Mathematica provides functionality for unsupervised anomaly detection across multiple data types
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- Wolfram Mathematica provides functionality for unsupervised anomaly detection across multiple data types
## Datasets
- Anomaly detection benchmark data repository with carefully chosen data sets of the Ludwig-Maximilians-Universität München; Mirror at University of São Paulo.
- ODDS – ODDS: A large collection of publicly available outlier detection datasets with ground truth in different domains.
- Unsupervised Anomaly Detection Benchmark at Harvard Dataverse: Datasets for Unsupervised Anomaly Detection with ground truth.
- KMASH Data Repository at Research Data Australia having more than 12,000 anomaly detection datasets with ground truth.
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Software design is the process of conceptualizing how a software system will work before it is implemented or modified.
Software design also refers to the direct result of the design process the concepts of how the software will work which consists of both design documentation and undocumented concepts.
Software design usually is directed by goals for the resulting system and involves problem-solving and planning including both
high-level software architecture and low-level component and algorithm design.
In terms of the waterfall development process, software design is the activity of following requirements specification and before coding.
## General process
The design process enables a designer to model various aspects of a software system before it exists.
Creativity, past experience, a sense of what makes "good" software, and a commitment to quality are success factors for a competent design. However, the design process is not always a straightforward procedure.
The software design model can be compared to an architected plan for a house. High-level plans represent the totality of the house (e.g., a three-dimensional rendering of the house). Lower-level plans provide guidance for constructing each detail (e.g., the plumbing lay). Similarly, the software design model provides a variety of views of the proposed software solution.
## Iterative Design for Software Components
Software systems inherently deal with uncertainties, and the size of software components can significantly influence a system's outcomes, both positively and negatively.
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Similarly, the software design model provides a variety of views of the proposed software solution.
## Iterative Design for Software Components
Software systems inherently deal with uncertainties, and the size of software components can significantly influence a system's outcomes, both positively and negatively. Neal Ford and Mark Richards propose an iterative approach to address the challenge of identifying and right-sizing components. This method emphasizes continuous refinement as teams develop a more nuanced understanding of system behavior and requirements.
The approach typically involves a cycle with several stages:
- A high-level partitioning strategy is established, often categorized as technical or domain-based. Guidelines for the smallest meaningful deployable unit, referred to as "quanta," are defined. While these foundational decisions are made early, they may be revisited later in the cycle if necessary.
- Initial components are identified based on the established strategy.
- Requirements are assigned to the identified components.
- The roles and responsibilities of each component are analyzed to ensure clarity and minimize overlap.
- Architectural characteristics, such as scalability, fault tolerance, and maintainability, are evaluated.
- Components may be restructured based on feedback from development teams.
This cycle serves as a general framework and can be adapted to different domains.
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- Requirements are assigned to the identified components.
- The roles and responsibilities of each component are analyzed to ensure clarity and minimize overlap.
- Architectural characteristics, such as scalability, fault tolerance, and maintainability, are evaluated.
- Components may be restructured based on feedback from development teams.
This cycle serves as a general framework and can be adapted to different domains.
## Value
Software design documentation may be reviewed or presented to allow constraints, specifications and even requirements to be adjusted prior to coding. Redesign may occur after a review of a programmed simulation or prototype. It is possible to design software in the process of coding, without a plan or requirement analysis, but for more complex projects this is less feasible. A separate design prior to coding allows for multidisciplinary designers and subject-matter experts (SMEs) to collaborate with programmers in order to produce software that is useful and technically sound.
## Requirements analysis
One component of software design is software requirements analysis (SRA). SRA is a part of the software development process that lists specifications used in software engineering.
The output of the analysis is smaller problems to solve.
In contrast, the design focuses on capabilities, and thus multiple designs for the same problem can exist. Depending on the environment, the design often varies, whether it is created from reliable frameworks or implemented with suitable design patterns.
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In contrast, the design focuses on capabilities, and thus multiple designs for the same problem can exist. Depending on the environment, the design often varies, whether it is created from reliable frameworks or implemented with suitable design patterns.
## Artifacts
A design process may include the production of artifacts such as flow chart, use case, Pseudocode, Unified Modeling Language model and other Fundamental modeling concepts. For user centered software, design may involve user experience design yielding a storyboard to help determine those specifications.
Sometimes the output of a design process is
design documentation.
## Design principles
Basic design principles enable a software engineer to navigate the design process. Davis suggests a set of principles for software design, which have been adapted and extended in the following list:
- The design process should not suffer from "tunnel vision". A good designer should consider alternative approaches, judging each based on the requirements of the problem, the resources available to do the job.
- The design should be traceable to the analysis model. Because a single element of the design model can often be traced back to multiple requirements, it is necessary to have a means for tracking how requirements have been satisfied by the design model.
- The design should not reinvent the wheel. Systems are constructed using a set of design patterns, many of which have likely been encountered before. These patterns should always be chosen as an alternative to reinvention.
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Systems are constructed using a set of design patterns, many of which have likely been encountered before. These patterns should always be chosen as an alternative to reinvention. Time is short and resources are limited; design time should be invested in representing (truly new) ideas by integrating patterns that already exist (when applicable).
- The design should "minimize the intellectual distance" between the software and the problem as it exists in the real world. That is, the structure of the software design should, whenever possible, mimic the structure of the problem domain.
- The design should exhibit uniformity and integration. A design is uniform if it appears fully coherent. In order to achieve this outcome, rules of style and format should be defined for a design team before design work begins. A design is integrated if care is taken in defining interfaces between design components.
- The design should be structured to accommodate change. The design concepts discussed in the next section enable a design to achieve this principle.
- The design should be structured to degrade gently, even when aberrant data, events, or operating conditions are encountered. Well-designed software should never "bomb"; it should be designed to accommodate unusual circumstances, and if it must terminate processing, it should do so in a graceful manner.
- Design is not coding, coding is not design.
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Well-designed software should never "bomb"; it should be designed to accommodate unusual circumstances, and if it must terminate processing, it should do so in a graceful manner.
- Design is not coding, coding is not design. Even when detailed procedural designs are created for program components, the level of abstraction of the design model is higher than the source code. The only design decisions made at the coding level should address the small implementation details that enable the procedural design to be coded.
- The design should be assessed for quality as it is being created, not after the fact. A variety of design concepts and design measures are available to assist the designer in assessing quality throughout the development process.
- The design should be reviewed to minimize conceptual (semantic) errors. There is sometimes a tendency to focus on minutiae when the design is reviewed, missing the forest for the trees. A design team should ensure that major conceptual elements of the design (omissions, ambiguity, inconsistency) have been addressed before worrying about the syntax of the design model.
## Design concepts
Design concepts provide a designer with a foundation from which more sophisticated methods can be applied. A set of design concepts has evolved including:
- Abstraction - Abstraction is the process or result of generalization by reducing the information content of a concept or an observable phenomenon, typically to retain only information that is relevant for a particular purpose.
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## Design concepts
Design concepts provide a designer with a foundation from which more sophisticated methods can be applied. A set of design concepts has evolved including:
- Abstraction - Abstraction is the process or result of generalization by reducing the information content of a concept or an observable phenomenon, typically to retain only information that is relevant for a particular purpose. It is an act of Representing essential features without including the background details or explanations.
- Refinement - It is the process of elaboration. A hierarchy is developed by decomposing a macroscopic statement of function in a step-wise fashion until programming language statements are reached. In each step, one or several instructions of a given program are decomposed into more detailed instructions. Abstraction and Refinement are complementary concepts.
- Modularity - Software architecture is divided into components called modules.
- Software Architecture - It refers to the overall structure of the software and the ways in which that structure provides conceptual integrity for a system. Good software architecture will yield a good return on investment with respect to the desired outcome of the project, e.g. in terms of performance, quality, schedule and cost.
- Control Hierarchy - A program structure that represents the organization of a program component and implies a hierarchy of control.
- Structural Partitioning - The program structure can be divided horizontally and vertically.
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- Control Hierarchy - A program structure that represents the organization of a program component and implies a hierarchy of control.
- Structural Partitioning - The program structure can be divided horizontally and vertically. Horizontal partitions define separate branches of modular hierarchy for each major program function. Vertical partitioning suggests that control and work should be distributed top-down in the program structure.
- Data Structure - It is a representation of the logical relationship among individual elements of data.
- Software Procedure - It focuses on the processing of each module individually.
- Information Hiding - Modules should be specified and designed so that information contained within a module is inaccessible to other modules that have no need for such information.
In his object model, Grady Booch mentions Abstraction, Encapsulation, Modularisation, and Hierarchy as fundamental software design principles. The acronym PHAME (Principles of Hierarchy, Abstraction, Modularisation, and Encapsulation) is sometimes used to refer to these four fundamental principles.
## Design considerations
There are many aspects to consider in the design of a piece of software. The importance of each consideration should reflect the goals and expectations that the software is being created to meet. Some of these aspects are:
- Compatibility - The software is able to operate with other products that are designed for interoperability with another product.
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The importance of each consideration should reflect the goals and expectations that the software is being created to meet. Some of these aspects are:
- Compatibility - The software is able to operate with other products that are designed for interoperability with another product. For example, a piece of software may be backward-compatible with an older version of itself.
- Extensibility - New capabilities can be added to the software without major changes to the underlying architecture.
- Modularity - the resulting software comprises well defined, independent components which leads to better maintainability. The components could be then implemented and tested in isolation before being integrated to form a desired software system. This allows division of work in a software development project.
- Fault-tolerance - The software is resistant to and able to recover from component failure.
- Maintainability - A measure of how easily bug fixes or functional modifications can be accomplished. High maintainability can be the product of modularity and extensibility.
- Reliability (Software durability) - The software is able to perform a required function under stated conditions for a specified period of time.
- Reusability - The ability to use some or all of the aspects of the preexisting software in other projects with little to no modification.
- Robustness - The software is able to operate under stress or tolerate unpredictable or invalid input.
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The ability to use some or all of the aspects of the preexisting software in other projects with little to no modification.
- Robustness - The software is able to operate under stress or tolerate unpredictable or invalid input. For example, it can be designed with resilience to low memory conditions.
- Security - The software is able to withstand and resist hostile acts and influences.
- Usability - The software user interface must be usable for its target user/audience. Default values for the parameters must be chosen so that they are a good choice for the majority of the users.
- Performance - The software performs its tasks within a time-frame that is acceptable for the user, and does not require too much memory.
- Portability - The software should be usable across a number of different conditions and environments.
- Scalability - The software adapts well to increasing data or added features or number of users. According to Marc Brooker: "a system is scalable in the range where marginal cost of additional workload is nearly constant." Serverless technologies fit this definition but you need to consider total cost of ownership not just the infra cost.
## Modeling language
A modeling language can be used to express information, knowledge or systems in a structure that is defined by a consistent set of rules. These rules are used for interpretation of the components within the structure. A modeling language can be graphical or textual.
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These rules are used for interpretation of the components within the structure. A modeling language can be graphical or textual. Examples of graphical modeling languages for software design include:
- Architecture description language (ADL) is a language used to describe and represent the software architecture of a software system.
- Business Process Modeling Notation (BPMN) is an example of a Process Modeling language.
- EXPRESS and EXPRESS-G (ISO 10303-11) is an international standard general-purpose data modeling language.
- Extended Enterprise Modeling Language (EEML) is commonly used for business process modeling across a number of layers.
- Flowcharts are schematic representations of algorithms or other step-wise processes.
- Fundamental Modeling Concepts (FMC) is modeling language for software-intensive systems.
- IDEF is a family of modeling languages, the most notable of which include IDEF0 for functional modeling, IDEF1X for information modeling, and IDEF5 for modeling ontologies.
- Jackson Structured Programming (JSP) is a method for structured programming based on correspondences between data stream structure and program structure.
- LePUS3 is an object-oriented visual Design Description Language and a formal specification language that is suitable primarily for modeling large object-oriented (Java, C++, C#) programs and design patterns.
- Unified Modeling Language (UML) is a general modeling language to describe software both structurally and behaviorally.
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- Extended Enterprise Modeling Language (EEML) is commonly used for business process modeling across a number of layers.
- Flowcharts are schematic representations of algorithms or other step-wise processes.
- Fundamental Modeling Concepts (FMC) is modeling language for software-intensive systems.
- IDEF is a family of modeling languages, the most notable of which include IDEF0 for functional modeling, IDEF1X for information modeling, and IDEF5 for modeling ontologies.
- Jackson Structured Programming (JSP) is a method for structured programming based on correspondences between data stream structure and program structure.
- LePUS3 is an object-oriented visual Design Description Language and a formal specification language that is suitable primarily for modeling large object-oriented (Java, C++, C#) programs and design patterns.
- Unified Modeling Language (UML) is a general modeling language to describe software both structurally and behaviorally. It has a graphical notation and allows for extension with a Profile (UML).
- Alloy (specification language) is a general purpose specification language for expressing complex structural constraints and behavior in a software system. It provides a concise language base on first-order relational logic.
- Systems Modeling Language (SysML) is a new general-purpose modeling language for systems engineering.
- Service-oriented modeling framework (SOMF)
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It provides a concise language base on first-order relational logic.
- Systems Modeling Language (SysML) is a new general-purpose modeling language for systems engineering.
- Service-oriented modeling framework (SOMF)
## Design patterns
A software designer may identify a design aspect which has been visited and perhaps even solved by others in the past. A template or pattern describing a solution to a common problem is known as a design pattern. The reuse of such patterns can increase software development velocity.
## Code as design
The difficulty of using the term "design" in relation to software is that in some senses, the source code of a program is the design for the program that it produces. To the extent that this is true, "software design" refers to the design of the design. Edsger W. Dijkstra referred to this layering of semantic levels as the "radical novelty" of computer programming, and Donald Knuth used his experience writing TeX to describe the futility of attempting to design a program prior to implementing it:
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In mathematics, a function from a set to a set assigns to each element of exactly one element of . The set is called the domain of the function and the set is called the codomain of the function.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept.
A function is often denoted by a letter such as , or . The value of a function at an element of its domain (that is, the element of the codomain that is associated with ) is denoted by ; for example, the value of at is denoted by .
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A function is often denoted by a letter such as , or . The value of a function at an element of its domain (that is, the element of the codomain that is associated with ) is denoted by ; for example, the value of at is denoted by . Commonly, a specific function is defined by means of an expression depending on , such as
$$
f(x)=x^2+1;
$$
in this case, some computation, called , may be needed for deducing the value of the function at a particular value; for example, if
$$
f(x)=x^2+1,
$$
then
$$
f(4)=4^2+1=17.
$$
Given its domain and its codomain, a function is uniquely represented by the set of all pairs , called the graph of the function, a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.
Functions are widely used in science, engineering, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.
The concept of a function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See History of the function concept for details.
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The concept of a function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See History of the function concept for details.
## Definition
A function from a set to a set is an assignment of one element of to each element of . The set is called the domain of the function and the set is called the codomain of the function.
If the element in is assigned to in by the function , one says that maps to , and this is commonly written
$$
y=f(x).
$$
In this notation, is the argument or variable of the function.
A specific element of is a value of the variable, and the corresponding element of is the value of the function at , or the image of under the function. The image of a function, sometimes called its range, is the set of the images of all elements in the domain.
A function , its domain , and its codomain are often specified by the notation _ BLOCK1_ One may write
$$
x\mapsto y
$$
instead of
$$
y=f(x)
$$
, where the symbol
$$
\mapsto
$$
(read 'maps to') is used to specify where a particular element in the domain is mapped to by . This allows the definition of a function without naming. For example, the square function is the function
$$
x\mapsto x^2.
$$
The
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This allows the definition of a function without naming. For example, the square function is the function
$$
x\mapsto x^2.
$$
The domain and codomain are not always explicitly given when a function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if
$$
f:\R\to\R
$$
is a real function, the determination of the domain of the function
$$
x\mapsto 1/f(x)
$$
requires knowing the zeros of This is one of the reasons for which, in mathematical analysis, "a function may refer to a function having a proper subset of as a domain. For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable whose domain is a proper subset of the real numbers, typically a subset that contains a non-empty open interval. Such a function is then called a partial function.
A function on a set means a function from the domain , without specifying a codomain. However, some authors use it as shorthand for saying that the function is .
### Formal definition
The above definition of a function is essentially that of the founders of calculus, Leibniz, Newton and Euler.
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However, some authors use it as shorthand for saying that the function is .
### Formal definition
The above definition of a function is essentially that of the founders of calculus, Leibniz, Newton and Euler. However, it cannot be formalized, since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of set theory. This set-theoretic definition is based on the fact that a function establishes a relation between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, a binary relation between two sets and is a subset of the set of all ordered pairs
$$
(x, y)
$$
such that
$$
x\in X
$$
and
$$
y\in Y.
$$
The set of all these pairs is called the Cartesian product of and and denoted
$$
X\times Y.
$$
Thus, the above definition may be formalized as follows.
A function with domain and codomain is a binary relation between and that satisfies the two following conditions:
- For every
$$
x
$$
in
$$
X
$$
there exists
$$
y
$$
in
$$
Y
$$
such that _ BLOCK8_-
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A function with domain and codomain is a binary relation between and that satisfies the two following conditions:
- For every
$$
x
$$
in
$$
X
$$
there exists
$$
y
$$
in
$$
Y
$$
such that _ BLOCK8_- If
$$
(x,y)\in R
$$
and
$$
(x,z)\in R,
$$
then _ BLOCK11_This definition may be rewritten more formally, without referring explicitly to the concept of a relation, but using more notation (including set-builder notation):
A function is formed by three sets, the domain
$$
X,
$$
the codomain
$$
Y,
$$
and the graph
$$
R
$$
that satisfy the three following conditions.
-
$$
R \subseteq \{(x,y) \mid x\in X, y\in Y\}
$$
-
$$
\forall x\in X, \exists y\in Y, \left(x, y\right) \in R \qquad
$$
-
$$
(x,y)\in R \land (x,z)\in R \implies y=z\qquad
$$
### Partial functions
Partial functions are defined similarly to ordinary functions, with the "total" condition removed.
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BLOCK11_This definition may be rewritten more formally, without referring explicitly to the concept of a relation, but using more notation (including set-builder notation):
A function is formed by three sets, the domain
$$
X,
$$
the codomain
$$
Y,
$$
and the graph
$$
R
$$
that satisfy the three following conditions.
-
$$
R \subseteq \{(x,y) \mid x\in X, y\in Y\}
$$
-
$$
\forall x\in X, \exists y\in Y, \left(x, y\right) \in R \qquad
$$
-
$$
(x,y)\in R \land (x,z)\in R \implies y=z\qquad
$$
### Partial functions
Partial functions are defined similarly to ordinary functions, with the "total" condition removed. That is, a partial function from to is a binary relation between and such that, for every
$$
x\in X,
$$
there is at most one in such that
$$
(x,y) \in R.
$$
Using functional notation, this means that, given
$$
x\in X,
$$
either _ BLOCK3_ is in , or it is undefined.
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That is, a partial function from to is a binary relation between and such that, for every
$$
x\in X,
$$
there is at most one in such that
$$
(x,y) \in R.
$$
Using functional notation, this means that, given
$$
x\in X,
$$
either _ BLOCK3_ is in , or it is undefined.
The set of the elements of such that
$$
f(x)
$$
is defined and belongs to is called the domain of definition of the function. A partial function from to is thus a ordinary function that has as its domain a subset of called the domain of definition of the function. If the domain of definition equals , one often says that the partial function is a total function.
In several areas of mathematics the term "function" refers to partial functions rather than to ordinary functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain.
## In calculus
, a real-valued function of a real variable or real function is a partial function from the set
$$
\R
$$
of the real numbers to itself. Given a real function
$$
f:x\mapsto f(x)
$$
its multiplicative inverse
$$
x\mapsto 1/f(x)
$$
is also a real function.
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In calculus
, a real-valued function of a real variable or real function is a partial function from the set
$$
\R
$$
of the real numbers to itself. Given a real function
$$
f:x\mapsto f(x)
$$
its multiplicative inverse
$$
x\mapsto 1/f(x)
$$
is also a real function. The determination of the domain of definition of a multiplicative inverse of a (partial) function amounts to compute the zeros of the function, the values where the function is defined but not its multiplicative inverse.
Similarly, a function of a complex variable is generally a partial function with a domain of definition included in the set
$$
\Complex
$$
of the complex numbers. The difficulty of determining the domain of definition of a complex function is illustrated by the multiplicative inverse of the Riemann zeta function: the determination of the domain of definition of the function
$$
z\mapsto 1/\zeta(z)
$$
is more or less equivalent to the proof or disproof of one of the major open problems in mathematics, the Riemann hypothesis.
In computability theory, a general recursive function is a partial function from the integers to the integers whose values can be computed by an algorithm (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever.
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In computability theory, a general recursive function is a partial function from the integers to the integers whose values can be computed by an algorithm (roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether belongs to its domain of definition (see Halting problem).
### Multivariate functions
A multivariate function, multivariable function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed.
Formally, a function of variables is a function whose domain is a set of -tuples. For example, multiplication of integers is a function of two variables, or bivariate function, whose domain is the set of all ordered pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every binary operation. The graph of a bivariate surface over a two-dimensional real domain may be interpreted as defining a parametric surface, as used in, e.g., bivariate interpolation.
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The same is true for every binary operation. The graph of a bivariate surface over a two-dimensional real domain may be interpreted as defining a parametric surface, as used in, e.g., bivariate interpolation.
Commonly, an -tuple is denoted enclosed between parentheses, such as in
$$
(1,2,\ldots, n).
$$
When using functional notation, one usually omits the parentheses surrounding tuples, writing
$$
f(x_1,\ldots,x_n)
$$
instead of
$$
f((x_1,\ldots,x_n)).
$$
Given sets
$$
X_1,\ldots, X_n,
$$
the set of all -tuples
$$
(x_1,\ldots,x_n)
$$
such that
$$
x_1\in X_1, \ldots, x_n\in X_n
$$
is called the Cartesian product of
$$
X_1,\ldots, X_n,
$$
and denoted
$$
X_1\times\cdots\times X_n.
$$
Therefore, a multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain.
$$
f: U\to Y,
$$
where the domain has the form
$$
U\subseteq X_1\times\cdots\times X_n.
$$
If all the
$$
X_i
$$
are equal to the set
$$
\R
$$
of the real numbers or to the set
$$
\C
$$
of the complex numbers, one talks respectively of a function of several real variables or of a function of several complex variables.
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The graph of a bivariate surface over a two-dimensional real domain may be interpreted as defining a parametric surface, as used in, e.g., bivariate interpolation.
Commonly, an -tuple is denoted enclosed between parentheses, such as in
$$
(1,2,\ldots, n).
$$
When using functional notation, one usually omits the parentheses surrounding tuples, writing
$$
f(x_1,\ldots,x_n)
$$
instead of
$$
f((x_1,\ldots,x_n)).
$$
Given sets
$$
X_1,\ldots, X_n,
$$
the set of all -tuples
$$
(x_1,\ldots,x_n)
$$
such that
$$
x_1\in X_1, \ldots, x_n\in X_n
$$
is called the Cartesian product of
$$
X_1,\ldots, X_n,
$$
and denoted
$$
X_1\times\cdots\times X_n.
$$
Therefore, a multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain.
$$
f: U\to Y,
$$
where the domain has the form
$$
U\subseteq X_1\times\cdots\times X_n.
$$
If all the
$$
X_i
$$
are equal to the set
$$
\R
$$
of the real numbers or to the set
$$
\C
$$
of the complex numbers, one talks respectively of a function of several real variables or of a function of several complex variables.
## Notation
There are various standard ways for denoting functions.
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Commonly, an -tuple is denoted enclosed between parentheses, such as in
$$
(1,2,\ldots, n).
$$
When using functional notation, one usually omits the parentheses surrounding tuples, writing
$$
f(x_1,\ldots,x_n)
$$
instead of
$$
f((x_1,\ldots,x_n)).
$$
Given sets
$$
X_1,\ldots, X_n,
$$
the set of all -tuples
$$
(x_1,\ldots,x_n)
$$
such that
$$
x_1\in X_1, \ldots, x_n\in X_n
$$
is called the Cartesian product of
$$
X_1,\ldots, X_n,
$$
and denoted
$$
X_1\times\cdots\times X_n.
$$
Therefore, a multivariate function is a function that has a Cartesian product or a proper subset of a Cartesian product as a domain.
$$
f: U\to Y,
$$
where the domain has the form
$$
U\subseteq X_1\times\cdots\times X_n.
$$
If all the
$$
X_i
$$
are equal to the set
$$
\R
$$
of the real numbers or to the set
$$
\C
$$
of the complex numbers, one talks respectively of a function of several real variables or of a function of several complex variables.
## Notation
There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below.
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## Notation
There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below.
### Functional notation
The functional notation requires that a name is given to the function, which, in the case of a unspecified function is often the letter . Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in
$$
f(x), \quad \sin(3),\quad \text{or}\quad f(x^2+1).
$$
The argument between the parentheses may be a variable, often , that represents an arbitrary element of the domain of the function, a specific element of the domain ( in the above example), or an expression that can be evaluated to an element of the domain (
$$
x^2+1
$$
in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let
$$
f(x)=\sin(x^2+1)
$$
".
When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write instead of .
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When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write instead of .
Functional notation was first used by Leonhard Euler in 1734. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case, a roman type is customarily used instead, such as "" for the sine function, in contrast to italic font for single-letter symbols.
The functional notation is often used colloquially for referring to a function and simultaneously naming its argument, such as in "let
$$
f(x)
$$
be a function". This is an abuse of notation that is useful for a simpler formulation.
### Arrow notation
Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. It uses the ↦ arrow symbol, pronounced "maps to". For example,
$$
x\mapsto x+1
$$
is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of
$$
\R
$$
is implied.
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For example,
$$
x\mapsto x+1
$$
is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of
$$
\R
$$
is implied.
The domain and codomain can also be explicitly stated, for example:
$$
\begin{align}
\operatorname{sqr}\colon \Z &\to \Z\\
x &\mapsto x^2.\end{align}
$$
This defines a function from the integers to the integers that returns the square of its input.
As a common application of the arrow notation, suppose
$$
f: X\times X\to Y;\;(x,t) \mapsto f(x,t)
$$
is a function in two variables, and we want to refer to a partially applied function
$$
X\to Y
$$
produced by fixing the second argument to the value without introducing a new function name. The map in question could be denoted
$$
x\mapsto f(x,t_0)
$$
using the arrow notation. The expression
$$
x\mapsto f(x,t_0)
$$
(read: "the map taking to of comma nought") represents this new function with just one argument, whereas the expression refers to the value of the function at the
### Index notation
Index notation may be used instead of functional notation.
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The expression
$$
x\mapsto f(x,t_0)
$$
(read: "the map taking to of comma nought") represents this new function with just one argument, whereas the expression refers to the value of the function at the
### Index notation
Index notation may be used instead of functional notation. That is, instead of writing , one writes
$$
f_x.
$$
This is typically the case for functions whose domain is the set of the natural numbers. Such a function is called a sequence, and, in this case the element
$$
f_n
$$
is called the th element of the sequence.
The index notation can also be used for distinguishing some variables called parameters from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map
$$
x\mapsto f(x,t)
$$
(see above) would be denoted
$$
f_t
$$
using index notation, if we define the collection of maps
$$
f_t
$$
by the formula
$$
f_t(x)=f(x,t)
$$
for all
$$
x,t\in X
$$
.
### Dot notation
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For example, the map
$$
x\mapsto f(x,t)
$$
(see above) would be denoted
$$
f_t
$$
using index notation, if we define the collection of maps
$$
f_t
$$
by the formula
$$
f_t(x)=f(x,t)
$$
for all
$$
x,t\in X
$$
.
### Dot notation
In the notation
$$
x\mapsto f(x),
$$
the symbol does not represent any value; it is simply a placeholder, meaning that, if is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, may be replaced by any symbol, often an interpunct "". This may be useful for distinguishing the function from its value at .
For example,
$$
a(\cdot)^2
$$
may stand for the function
$$
x\mapsto ax^2
$$
, and
$$
\int_a^{\, (\cdot)} f(u)\,du
$$
may stand for a function defined by an integral with variable upper bound:
$$
x\mapsto \int_a^x f(u)\,du
$$
.
### Specialized notations
There are other, specialized notations for functions in sub-disciplines of mathematics.
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For example,
$$
a(\cdot)^2
$$
may stand for the function
$$
x\mapsto ax^2
$$
, and
$$
\int_a^{\, (\cdot)} f(u)\,du
$$
may stand for a function defined by an integral with variable upper bound:
$$
x\mapsto \int_a^x f(u)\,du
$$
.
### Specialized notations
There are other, specialized notations for functions in sub-disciplines of mathematics. For example, in linear algebra and functional analysis, linear forms and the vectors they act upon are denoted using a dual pair to show the underlying duality. This is similar to the use of bra–ket notation in quantum mechanics. In logic and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application. In category theory and homological algebra, networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.
### Functions of more than one variable
In some cases the argument of a function may be an ordered pair of elements taken from some set or sets.
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In category theory and homological algebra, networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.
### Functions of more than one variable
In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function can be defined as mapping any pair of real numbers
$$
(x, y)
$$
to the sum of their squares,
$$
x^2 + y^2
$$
. Such a function is commonly written as
$$
f(x, y)=x^2 + y^2
$$
and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as
$$
f(w,x, y)
$$
,
$$
f(w,x, y, z)
$$
.
## Other terms
TermDistinction from "function"Map/MappingNone; the terms are synonymous. A map can have any set as its codomain, while, in some contexts, typically in older books, the codomain of a function is specifically the set of real or complex numbers. Alternatively, a map is associated with a special structure (e.g. by explicitly specifying a structured codomain in its definition). For example, a linear map.
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Alternatively, a map is associated with a special structure (e.g. by explicitly specifying a structured codomain in its definition). For example, a linear map. HomomorphismA function between two structures of the same type that preserves the operations of the structure (e.g. a group homomorphism).MorphismA generalisation of homomorphisms to any category, even when the objects of the category are not sets (for example, a group defines a category with only one object, which has the elements of the group as morphisms; see for this example and other similar ones).
A function may also be called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g. maps of manifolds). In particular map may be used in place of homomorphism for the sake of succinctness (e.g., linear map or map from to instead of group homomorphism from to ). Some authors reserve the word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function.
Some authors, such as Serge Lang, use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term mapping for more general functions.
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Some authors reserve the word mapping for the case where the structure of the codomain belongs explicitly to the definition of the function.
Some authors, such as Serge Lang, use "function" only to refer to maps for which the codomain is a subset of the real or complex numbers, and use the term mapping for more general functions.
In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.
Whichever definition of map is used, related terms like domain, codomain, injective, continuous have the same meaning as for a function.
## Specifying a function
Given a function
$$
f
$$
, by definition, to each element
$$
x
$$
of the domain of the function
$$
f
$$
, there is a unique element associated to it, the value
$$
f(x)
$$
of
$$
f
$$
at
$$
x
$$
. There are several ways to specify or describe how _ BLOCK6_ is related to
$$
f(x)
$$
, both explicitly and implicitly. Sometimes, a theorem or an axiom asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function
$$
f
$$
.
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Sometimes, a theorem or an axiom asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function
$$
f
$$
.
### By listing function values
On a finite set a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. For example, if
$$
A = \{ 1, 2, 3 \}
$$
, then one can define a function
$$
f: A \to \mathbb{R}
$$
by
$$
f(1) = 2, f(2) = 3, f(3) = 4.
$$
### By a formula
Functions are often defined by an expression that describes a combination of arithmetic operations and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain.
For example, in the above example,
$$
f
$$
can be defined by the formula
$$
f(n) = n+1
$$
, for
$$
n\in\{1,2,3\}
$$
.
When a function is defined this way, the determination of its domain is sometimes difficult.
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For example, in the above example,
$$
f
$$
can be defined by the formula
$$
f(n) = n+1
$$
, for
$$
n\in\{1,2,3\}
$$
.
When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the zeros of auxiliary functions. Similarly, if square roots occur in the definition of a function from
$$
\mathbb{R}
$$
to
$$
\mathbb{R},
$$
the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative.
For example,
$$
f(x)=\sqrt{1+x^2}
$$
defines a function
$$
f: \mathbb{R} \to \mathbb{R}
$$
whose domain is
$$
\mathbb{R},
$$
because
$$
1+x^2
$$
is always positive if is a real number. On the other hand,
$$
f(x)=\sqrt{1-x^2}
$$
defines a function from the reals to the reals whose domain is reduced to the interval .
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For example,
$$
f(x)=\sqrt{1+x^2}
$$
defines a function
$$
f: \mathbb{R} \to \mathbb{R}
$$
whose domain is
$$
\mathbb{R},
$$
because
$$
1+x^2
$$
is always positive if is a real number. On the other hand,
$$
f(x)=\sqrt{1-x^2}
$$
defines a function from the reals to the reals whose domain is reduced to the interval . (In old texts, such a domain was called the domain of definition of the function.)
Functions can be classified by the nature of formulas that define them:
- A quadratic function is a function that may be written
$$
f(x) = ax^2+bx+c,
$$
where are constants.
- More generally, a polynomial function is a function that can be defined by a formula involving only additions, subtractions, multiplications, and exponentiation to nonnegative integer powers. For example,
$$
f(x) = x^3-3x-1
$$
and
$$
f(x) = (x-1)(x^3+1) +2x^2 -1
$$
are polynomial functions of
$$
x
$$
.
-
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Functions can be classified by the nature of formulas that define them:
- A quadratic function is a function that may be written
$$
f(x) = ax^2+bx+c,
$$
where are constants.
- More generally, a polynomial function is a function that can be defined by a formula involving only additions, subtractions, multiplications, and exponentiation to nonnegative integer powers. For example,
$$
f(x) = x^3-3x-1
$$
and
$$
f(x) = (x-1)(x^3+1) +2x^2 -1
$$
are polynomial functions of
$$
x
$$
.
- A rational function is the same, with divisions also allowed, such as
$$
f(x) = \frac{x-1}{x+1},
$$
and
$$
f(x) = \frac 1{x+1}+\frac 3x-\frac 2{x-1}.
$$
- An algebraic function is the same, with th roots and roots of polynomials also allowed.
- An elementary function is the same, with logarithms and exponential functions allowed.
### Inverse and implicit functions
A function
$$
f : X\to Y,
$$
with domain and codomain , is bijective, if for every in , there is one and only one element in such that .
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- An elementary function is the same, with logarithms and exponential functions allowed.
### Inverse and implicit functions
A function
$$
f : X\to Y,
$$
with domain and codomain , is bijective, if for every in , there is one and only one element in such that . In this case, the inverse function of is the function
$$
f^{-1} : Y \to X
$$
that maps
$$
y\in Y
$$
to the element
$$
x\in X
$$
such that . For example, the natural logarithm is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the exponential function, that maps the real numbers onto the positive numbers.
If a function
$$
f: X\to Y
$$
is not bijective, it may occur that one can select subsets
$$
E\subseteq X
$$
and
$$
F\subseteq Y
$$
such that the restriction of to is a bijection from to , and has thus an inverse. The inverse trigonometric functions are defined this way. For example, the cosine function induces, by restriction, a bijection from the interval onto the interval , and its inverse function, called arccosine, maps onto . The other inverse trigonometric functions are defined similarly.
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For example, the cosine function induces, by restriction, a bijection from the interval onto the interval , and its inverse function, called arccosine, maps onto . The other inverse trigonometric functions are defined similarly.
More generally, given a binary relation between two sets and , let be a subset of such that, for every
$$
x\in E,
$$
there is some
$$
y\in Y
$$
such that . If one has a criterion allowing selecting such a for every
$$
x\in E,
$$
this defines a function
$$
f: E\to Y,
$$
called an implicit function, because it is implicitly defined by the relation .
For example, the equation of the unit circle
$$
x^2+y^2=1
$$
defines a relation on real numbers. If there are two possible values of , one positive and one negative. For , these two values become both equal to 0. Otherwise, there is no possible value of . This means that the equation defines two implicit functions with domain and respective codomains and .
In this example, the equation can be solved in , giving
$$
y=\pm \sqrt{1-x^2},
$$
but, in more complicated examples, this is impossible.
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This means that the equation defines two implicit functions with domain and respective codomains and .
In this example, the equation can be solved in , giving
$$
y=\pm \sqrt{1-x^2},
$$
but, in more complicated examples, this is impossible. For example, the relation
$$
y^5+y+x=0
$$
defines as an implicit function of , called the Bring radical, which has
$$
\mathbb R
$$
as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and th roots.
The implicit function theorem provides mild differentiability conditions for existence and uniqueness of an implicit function in the neighborhood of a point.
### Using differential calculus
Many functions can be defined as the antiderivative of another function. This is the case of the natural logarithm, which is the antiderivative of that is 0 for . Another common example is the error function.
More generally, many functions, including most special functions, can be defined as solutions of differential equations. The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for .
Power series can be used to define functions on the domain in which they converge.
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The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for .
Power series can be used to define functions on the domain in which they converge. For example, the exponential function is given by
$$
e^x = \sum_{n=0}^{\infty} {x^n \over n!}
$$
. However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its Taylor series in some interval, this power series allows immediately enlarging the domain to a subset of the complex numbers, the disc of convergence of the series. Then analytic continuation allows enlarging further the domain for including almost the whole complex plane. This process is the method that is generally used for defining the logarithm, the exponential and the trigonometric functions of a complex number.
### By recurrence
Functions whose domain are the nonnegative integers, known as sequences, are sometimes defined by recurrence relations.
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This process is the method that is generally used for defining the logarithm, the exponential and the trigonometric functions of a complex number.
### By recurrence
Functions whose domain are the nonnegative integers, known as sequences, are sometimes defined by recurrence relations.
The factorial function on the nonnegative integers (
$$
n\mapsto n!
$$
) is a basic example, as it can be defined by the recurrence relation
$$
n!=n(n-1)!\quad\text{for}\quad n>0,
$$
and the initial condition
$$
0!=1.
$$
## Representing a function
A graph is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.
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As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by bar charts.
### Graphs and plots
Given a function
$$
f : X\to Y,
$$
its graph is, formally, the set
$$
G=\{(x,f(x))\mid x\in X\}.
$$
In the frequent case where and are subsets of the real numbers (or may be identified with such subsets, e.g. intervals), an element
$$
(x,y)\in G
$$
may be identified with a point having coordinates in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a plot that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the graph of the function. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the square function
$$
x\mapsto x^2,
$$
consisting of all points with coordinates
$$
(x, x^2)
$$
for
$$
x\in \R,
$$
yields, when depicted in Cartesian coordinates, the well known parabola.
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Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the square function
$$
x\mapsto x^2,
$$
consisting of all points with coordinates
$$
(x, x^2)
$$
for
$$
x\in \R,
$$
yields, when depicted in Cartesian coordinates, the well known parabola. If the same quadratic function
$$
x\mapsto x^2,
$$
with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates
$$
(r,\theta) =(x,x^2),
$$
the plot obtained is Fermat's spiral.
### Tables
A function can be represented as a table of values. If the domain of a function is finite, then the function can be completely specified in this way. For example, the multiplication function
$$
f:\{1,\ldots,5\}^2 \to \mathbb{R}
$$
defined as
$$
f(x,y)=xy
$$
can be represented by the familiar multiplication table
1 2 3 4 5 1 1 2 3 4 5 2 2 4 6 8 10 3 3 6 9 12 15 4 4 8 12 16 20 5 5 10 15 20 25
On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain.
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For example, the multiplication function
$$
f:\{1,\ldots,5\}^2 \to \mathbb{R}
$$
defined as
$$
f(x,y)=xy
$$
can be represented by the familiar multiplication table
1 2 3 4 5 1 1 2 3 4 5 2 2 4 6 8 10 3 3 6 9 12 15 4 4 8 12 16 20 5 5 10 15 20 25
On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed, interpolation can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:
1.289 0.9605571.290 0.9608351.291 0.9611121.292 0.9613871.293 0.961662
Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.
### Bar chart
A bar chart can represent a function whose domain is a finite set, the natural numbers, or the integers.
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For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places:
1.289 0.9605571.290 0.9608351.291 0.9611121.292 0.9613871.293 0.961662
Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.
### Bar chart
A bar chart can represent a function whose domain is a finite set, the natural numbers, or the integers. In this case, an element of the domain is represented by an interval of the -axis, and the corresponding value of the function, , is represented by a rectangle whose base is the interval corresponding to and whose height is (possibly negative, in which case the bar extends below the -axis).
## General properties
This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.
### Standard functions
There are a number of standard functions that occur frequently:
- For every set , there is a unique function, called the , or empty map, from the empty set to . The graph of an empty function is the empty set. The existence of empty functions is needed both for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements.
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The graph of an empty function is the empty set. The existence of empty functions is needed both for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements. Under the usual set-theoretic definition of a function as an ordered triplet (or equivalent ones), there is exactly one empty function for each set, thus the empty function
$$
\varnothing \to X
$$
is not equal to
$$
\varnothing \to Y
$$
if and only if
$$
X\ne Y
$$
, although their graphs are both the empty set.
- For every set and every singleton set , there is a unique function from to , which maps every element of to . This is a surjection (see below) unless is the empty set.
- Given a function
$$
f: X\to Y,
$$
the canonical surjection of onto its image
$$
f(X)=\{f(x)\mid x\in X\}
$$
is the function from to that maps to .
- For every subset of a set , the inclusion map of into is the injective (see below) function that maps every element of to itself.
- The identity function on a set , often denoted by , is the inclusion of into itself.
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For every subset of a set , the inclusion map of into is the injective (see below) function that maps every element of to itself.
- The identity function on a set , often denoted by , is the inclusion of into itself.
### Function composition
Given two functions
$$
f: X\to Y
$$
and
$$
g: Y\to Z
$$
such that the domain of is the codomain of , their composition is the function
$$
g \circ f: X \rightarrow Z
$$
defined by
$$
(g \circ f)(x) = g(f(x)).
$$
That is, the value of
$$
g \circ f
$$
is obtained by first applying to to obtain and then applying to the result to obtain . In this notation, the function that is applied first is always written on the right.
The composition
$$
g\circ f
$$
is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both
$$
g \circ f
$$
and
$$
f \circ g
$$
satisfy these conditions, the composition is not necessarily commutative, that is, the functions
$$
g \circ f
$$
and
$$
f \circ g
$$
need not be equal, but may deliver different values for the same argument.
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The composition
$$
g\circ f
$$
is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both
$$
g \circ f
$$
and
$$
f \circ g
$$
satisfy these conditions, the composition is not necessarily commutative, that is, the functions
$$
g \circ f
$$
and
$$
f \circ g
$$
need not be equal, but may deliver different values for the same argument. For example, let and , then
$$
g(f(x))=x^2+1
$$
and
$$
f(g(x)) = (x+1)^2
$$
agree just for
$$
x=0.
$$
The function composition is associative in the sense that, if one of
$$
(h\circ g)\circ f
$$
and
$$
h\circ (g\circ f)
$$
is defined, then the other is also defined, and they are equal, that is,
$$
(h\circ g)\circ f = h\circ (g\circ f).
$$
Therefore, it is usual to just write
$$
h\circ g\circ f.
$$
The identity functions
$$
\operatorname{id}_X
$$
and
$$
\operatorname{id}_Y
$$
are respectively a right identity and a left identity for functions from to .
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For example, let and , then
$$
g(f(x))=x^2+1
$$
and
$$
f(g(x)) = (x+1)^2
$$
agree just for
$$
x=0.
$$
The function composition is associative in the sense that, if one of
$$
(h\circ g)\circ f
$$
and
$$
h\circ (g\circ f)
$$
is defined, then the other is also defined, and they are equal, that is,
$$
(h\circ g)\circ f = h\circ (g\circ f).
$$
Therefore, it is usual to just write
$$
h\circ g\circ f.
$$
The identity functions
$$
\operatorname{id}_X
$$
and
$$
\operatorname{id}_Y
$$
are respectively a right identity and a left identity for functions from to . That is, if is a function with domain , and codomain , one has
$$
f\circ \operatorname{id}_X = \operatorname{id}_Y \circ f = f.
$$
### Image and preimage
Let
$$
f: X\to Y.
$$
The image under of an element of the domain is .
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### Image and preimage
Let
$$
f: X\to Y.
$$
The image under of an element of the domain is . If is any subset of , then the image of under , denoted , is the subset of the codomain consisting of all images of elements of , that is,
$$
f(A)=\{f(x)\mid x\in A\}.
$$
The image of is the image of the whole domain, that is, . It is also called the range of , although the term range may also refer to the codomain.
On the other hand, the inverse image or preimage under of an element of the codomain is the set of all elements of the domain whose images under equal . In symbols, the preimage of is denoted by
$$
f^{-1}(y)
$$
and is given by the equation
$$
f^{-1}(y) = \{x \in X \mid f(x) = y\}.
$$
Likewise, the preimage of a subset of the codomain is the set of the preimages of the elements of , that is, it is the subset of the domain consisting of all elements of whose images belong to .
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On the other hand, the inverse image or preimage under of an element of the codomain is the set of all elements of the domain whose images under equal . In symbols, the preimage of is denoted by
$$
f^{-1}(y)
$$
and is given by the equation
$$
f^{-1}(y) = \{x \in X \mid f(x) = y\}.
$$
Likewise, the preimage of a subset of the codomain is the set of the preimages of the elements of , that is, it is the subset of the domain consisting of all elements of whose images belong to . It is denoted by
$$
f^{-1}(B)
$$
and is given by the equation
$$
f^{-1}(B) = \{x \in X \mid f(x) \in B\}.
$$
For example, the preimage of
$$
\{4, 9\}
$$
under the square function is the set
$$
\{-3,-2,2,3\}
$$
.
By definition of a function, the image of an element of the domain is always a single element of the codomain. However, the preimage
$$
f^{-1}(y)
$$
of an element of the codomain may be empty or contain any number of elements.
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By definition of a function, the image of an element of the domain is always a single element of the codomain. However, the preimage
$$
f^{-1}(y)
$$
of an element of the codomain may be empty or contain any number of elements. For example, if is the function from the integers to themselves that maps every integer to 0, then
$$
f^{-1}(0) = \mathbb{Z}
$$
.
If
$$
f : X\to Y
$$
is a function, and are subsets of , and and are subsets of , then one has the following properties:
-
$$
A\subseteq B \Longrightarrow f(A)\subseteq f(B)
$$
-
$$
C\subseteq D \Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)
$$
-
$$
A \subseteq f^{-1}(f(A))
$$
-
$$
C \supseteq f(f^{-1}(C))
$$
-
$$
f(f^{-1}(f(A)))=f(A)
$$
-
$$
f^{-1}(f(f^{-1}(C)))=f^{-1}(C)
$$
The preimage by of an element of the codomain is sometimes called, in some contexts, the fiber of under .
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For example, if is the function from the integers to themselves that maps every integer to 0, then
$$
f^{-1}(0) = \mathbb{Z}
$$
.
If
$$
f : X\to Y
$$
is a function, and are subsets of , and and are subsets of , then one has the following properties:
-
$$
A\subseteq B \Longrightarrow f(A)\subseteq f(B)
$$
-
$$
C\subseteq D \Longrightarrow f^{-1}(C)\subseteq f^{-1}(D)
$$
-
$$
A \subseteq f^{-1}(f(A))
$$
-
$$
C \supseteq f(f^{-1}(C))
$$
-
$$
f(f^{-1}(f(A)))=f(A)
$$
-
$$
f^{-1}(f(f^{-1}(C)))=f^{-1}(C)
$$
The preimage by of an element of the codomain is sometimes called, in some contexts, the fiber of under .
If a function has an inverse (see below), this inverse is denoted
$$
f^{-1}.
$$
In this case
$$
f^{-1}(C)
$$
may denote either the image by
$$
f^{-1}
$$
or the preimage by of .
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If a function has an inverse (see below), this inverse is denoted
$$
f^{-1}.
$$
In this case
$$
f^{-1}(C)
$$
may denote either the image by
$$
f^{-1}
$$
or the preimage by of . This is not a problem, as these sets are equal. The notation
$$
f(A)
$$
and
$$
f^{-1}(C)
$$
may be ambiguous in the case of sets that contain some subsets as elements, such as
$$
\{x, \{x\}\}.
$$
In this case, some care may be needed, for example, by using square brackets
$$
f[A], f^{-1}[C]
$$
for images and preimages of subsets and ordinary parentheses for images and preimages of elements.
### Injective, surjective and bijective functions
Let
$$
f : X\to Y
$$
be a function.
The function is injective (or one-to-one, or is an injection) if for every two different elements and of . Equivalently, is injective if and only if, for every
$$
y\in Y,
$$
the preimage
$$
f^{-1}(y)
$$
contains at most one element. An empty function is always injective.
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Equivalently, is injective if and only if, for every
$$
y\in Y,
$$
the preimage
$$
f^{-1}(y)
$$
contains at most one element. An empty function is always injective. If is not the empty set, then is injective if and only if there exists a function
$$
g: Y\to X
$$
such that
$$
g\circ f=\operatorname{id}_X,
$$
that is, if has a left inverse. Proof: If is injective, for defining , one chooses an element
$$
x_0
$$
in (which exists as is supposed to be nonempty), and one defines by
$$
g(y)=x
$$
if
$$
y=f(x)
$$
and
$$
g(y)=x_0
$$
if
$$
y\not\in f(X).
$$
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that is, if has a left inverse. Proof: If is injective, for defining , one chooses an element
$$
x_0
$$
in (which exists as is supposed to be nonempty), and one defines by
$$
g(y)=x
$$
if
$$
y=f(x)
$$
and
$$
g(y)=x_0
$$
if
$$
y\not\in f(X).
$$
Conversely, if
$$
g\circ f=\operatorname{id}_X,
$$
and
$$
y=f(x),
$$
then
$$
x=g(y),
$$
and thus
$$
f^{-1}(y)=\{x\}.
$$
The function is surjective (or onto, or is a surjection) if its range
$$
f(X)
$$
equals its codomain
$$
Y
$$
, that is, if, for each element
$$
y
$$
of the codomain, there exists some element
$$
x
$$
of the domain such that
$$
f(x) = y
$$
(in other words, the preimage
$$
f^{-1}(y)
$$
of every
$$
y\in Y
$$
is nonempty).
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Proof: If is injective, for defining , one chooses an element
$$
x_0
$$
in (which exists as is supposed to be nonempty), and one defines by
$$
g(y)=x
$$
if
$$
y=f(x)
$$
and
$$
g(y)=x_0
$$
if
$$
y\not\in f(X).
$$
Conversely, if
$$
g\circ f=\operatorname{id}_X,
$$
and
$$
y=f(x),
$$
then
$$
x=g(y),
$$
and thus
$$
f^{-1}(y)=\{x\}.
$$
The function is surjective (or onto, or is a surjection) if its range
$$
f(X)
$$
equals its codomain
$$
Y
$$
, that is, if, for each element
$$
y
$$
of the codomain, there exists some element
$$
x
$$
of the domain such that
$$
f(x) = y
$$
(in other words, the preimage
$$
f^{-1}(y)
$$
of every
$$
y\in Y
$$
is nonempty). If, as usual in modern mathematics, the axiom of choice is assumed, then is surjective if and only if there exists a function
$$
g: Y\to X
$$
such that
$$
f\circ g=\operatorname{id}_Y,
$$
that is, if has a right inverse.
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Conversely, if
$$
g\circ f=\operatorname{id}_X,
$$
and
$$
y=f(x),
$$
then
$$
x=g(y),
$$
and thus
$$
f^{-1}(y)=\{x\}.
$$
The function is surjective (or onto, or is a surjection) if its range
$$
f(X)
$$
equals its codomain
$$
Y
$$
, that is, if, for each element
$$
y
$$
of the codomain, there exists some element
$$
x
$$
of the domain such that
$$
f(x) = y
$$
(in other words, the preimage
$$
f^{-1}(y)
$$
of every
$$
y\in Y
$$
is nonempty). If, as usual in modern mathematics, the axiom of choice is assumed, then is surjective if and only if there exists a function
$$
g: Y\to X
$$
such that
$$
f\circ g=\operatorname{id}_Y,
$$
that is, if has a right inverse. The axiom of choice is needed, because, if is surjective, one defines by
$$
g(y)=x,
$$
where
$$
x
$$
is an arbitrarily chosen element of _
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If, as usual in modern mathematics, the axiom of choice is assumed, then is surjective if and only if there exists a function
$$
g: Y\to X
$$
such that
$$
f\circ g=\operatorname{id}_Y,
$$
that is, if has a right inverse. The axiom of choice is needed, because, if is surjective, one defines by
$$
g(y)=x,
$$
where
$$
x
$$
is an arbitrarily chosen element of _ BLOCK25_The function is bijective (or is a bijection or a one-to-one correspondence) if it is both injective and surjective. That is, is bijective if, for every
$$
y\in Y,
$$
the preimage
$$
f^{-1}(y)
$$
contains exactly one element. The function is bijective if and only if it admits an inverse function, that is, a function
$$
g : Y\to X
$$
such that
$$
g\circ f=\operatorname{id}_X
$$
and
$$
f\circ g=\operatorname{id}_Y.
$$
(Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward).
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That is, is bijective if, for every
$$
y\in Y,
$$
the preimage
$$
f^{-1}(y)
$$
contains exactly one element. The function is bijective if and only if it admits an inverse function, that is, a function
$$
g : Y\to X
$$
such that
$$
g\circ f=\operatorname{id}_X
$$
and
$$
f\circ g=\operatorname{id}_Y.
$$
(Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward).
Every function
$$
f: X\to Y
$$
may be factorized as the composition
$$
i\circ s
$$
of a surjection followed by an injection, where is the canonical surjection of onto and is the canonical injection of into . This is the canonical factorization of .
"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function.
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"One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the Bourbaki group and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement " maps onto " differs from " maps into ", in that the former implies that is surjective, while the latter makes no assertion about the nature of . In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage of being more symmetrical.
### Restriction and extension
If
$$
f : X \to Y
$$
is a function and is a subset of , then the restriction of
$$
f
$$
to S, denoted
$$
f|_S
$$
, is the function from to defined by
$$
f|_S(x) = f(x)
$$
for all in .
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Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage of being more symmetrical.
### Restriction and extension
If
$$
f : X \to Y
$$
is a function and is a subset of , then the restriction of
$$
f
$$
to S, denoted
$$
f|_S
$$
, is the function from to defined by
$$
f|_S(x) = f(x)
$$
for all in . Restrictions can be used to define partial inverse functions: if there is a subset of the domain of a function
$$
f
$$
such that
$$
f|_S
$$
is injective, then the canonical surjection of
$$
f|_S
$$
onto its image _ BLOCK7_ is a bijection, and thus has an inverse function from
$$
f(S)
$$
to . One application is the definition of inverse trigonometric functions. For example, the cosine function is injective when restricted to the interval . The image of this restriction is the interval , and thus the restriction has an inverse function from to , which is called arccosine and is denoted .
Function restriction may also be used for "gluing" functions together. Let
$$
X=\bigcup_{i\in I}U_i
$$
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Function restriction may also be used for "gluing" functions together. Let
$$
X=\bigcup_{i\in I}U_i
$$
be the decomposition of as a union of subsets, and suppose that a function
$$
f_i : U_i \to Y
$$
is defined on each
$$
U_i
$$
such that for each pair
$$
i, j
$$
of indices, the restrictions of
$$
f_i
$$
and
$$
f_j
$$
to
$$
U_i \cap U_j
$$
are equal. Then this defines a unique function
$$
f : X \to Y
$$
such that
$$
f|_{U_i} = f_i
$$
for all . This is the way that functions on manifolds are defined.
An extension of a function is a function such that is a restriction of . A typical use of this concept is the process of analytic continuation, that allows extending functions whose domain is a small part of the complex plane to functions whose domain is almost the whole complex plane.
Here is another classical example of a function extension that is encountered when studying homographies of the real line. A homography is a function
$$
h(x)=\frac{ax+b}{cx+d}
$$
such that .
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Here is another classical example of a function extension that is encountered when studying homographies of the real line. A homography is a function
$$
h(x)=\frac{ax+b}{cx+d}
$$
such that . Its domain is the set of all real numbers different from
$$
-d/c,
$$
and its image is the set of all real numbers different from
$$
a/c.
$$
If one extends the real line to the projectively extended real line by including , one may extend to a bijection from the extended real line to itself by setting
$$
h(\infty)=a/c
$$
and
$$
h(-d/c)=\infty
$$
.
In calculus
The idea of function, starting in the 17th century, was fundamental to the new infinitesimal calculus. At that time, only real-valued functions of a real variable were considered, and all functions were assumed to be smooth. But the definition was soon extended to functions of several variables and to functions of a complex variable. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined.
Functions are now used throughout all areas of mathematics. In introductory calculus, when the word function is used without qualification, it means a real-valued function of a single real variable.
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Functions are now used throughout all areas of mathematics. In introductory calculus, when the word function is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with STEM majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as real analysis and complex analysis.
### Real function
A real function is a real-valued function of a real variable, that is, a function whose codomain is the field of real numbers and whose domain is a set of real numbers that contains an interval. In this section, these functions are simply called functions.
The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are continuous, differentiable, and even analytic. This regularity insures that these functions can be visualized by their graphs. In this section, all functions are differentiable in some interval.
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This regularity insures that these functions can be visualized by their graphs. In this section, all functions are differentiable in some interval.
Functions enjoy pointwise operations, that is, if and are functions, their sum, difference and product are functions defined by
$$
\begin{align}
(f+g)(x)&=f(x)+g(x)\\
(f-g)(x)&=f(x)-g(x)\\
(f\cdot g)(x)&=f(x)\cdot g(x)\\
\end{align}.
$$
The domains of the resulting functions are the intersection of the domains of and . The quotient of two functions is defined similarly by
$$
\frac fg(x)=\frac{f(x)}{g(x)},
$$
but the domain of the resulting function is obtained by removing the zeros of from the intersection of the domains of and .
The polynomial functions are defined by polynomials, and their domain is the whole set of real numbers. They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero.
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They include constant functions, linear functions and quadratic functions. Rational functions are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid division by zero. The simplest rational function is the function
$$
x\mapsto \frac 1x,
$$
whose graph is a hyperbola, and whose domain is the whole real line except for 0.
The derivative of a real differentiable function is a real function. An antiderivative of a continuous real function is a real function that has the original function as a derivative. For example, the function
$$
x\mapsto\frac 1x
$$
is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for , is a differentiable function called the natural logarithm.
A real function is monotonic in an interval if the sign of
$$
\frac{f(x)-f(y)}{x-y}
$$
does not depend of the choice of and in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function is monotonic in an interval , it has an inverse function, which is a real function with domain and image .
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If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function is monotonic in an interval , it has an inverse function, which is a real function with domain and image . This is how inverse trigonometric functions are defined in terms of trigonometric functions, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a bijection between the real numbers and the positive real numbers. This inverse is the exponential function.
Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of differential equations. For example, the sine and the cosine functions are the solutions of the linear differential equation
$$
y''+y=0
$$
such that
$$
\sin 0=0, \quad \cos 0=1, \quad\frac{\partial \sin x}{\partial x}(0)=1, \quad\frac{\partial \cos x}{\partial x}(0)=0.
$$
### Vector-valued function
When the elements of the codomain of a function are vectors, the function is said to be a vector-valued function.
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For example, the sine and the cosine functions are the solutions of the linear differential equation
$$
y''+y=0
$$
such that
$$
\sin 0=0, \quad \cos 0=1, \quad\frac{\partial \sin x}{\partial x}(0)=1, \quad\frac{\partial \cos x}{\partial x}(0)=0.
$$
### Vector-valued function
When the elements of the codomain of a function are vectors, the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its velocity vector is a vector-valued function.
Some vector-valued functions are defined on a subset of
$$
\mathbb{R}^n
$$
or other spaces that share geometric or topological properties of
$$
\mathbb{R}^n
$$
, such as manifolds. These vector-valued functions are given the name vector fields.
## Function space
In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space.
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These vector-valued functions are given the name vector fields.
## Function space
In mathematical analysis, and more specifically in functional analysis, a function space is a set of scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with a compact support (that is, they are zero outside some compact set) form a function space that is at the basis of the theory of distributions.
Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and topological properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or partial differential equations result of the study of function spaces.
## Multi-valued functions
Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a neighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point
$$
x_0,
$$
there are several possible starting values for the function.
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## Multi-valued functions
Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a neighbourhood of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point
$$
x_0,
$$
there are several possible starting values for the function.
For example, in defining the square root as the inverse function of the square function, for any positive real number
$$
x_0,
$$
there are two choices for the value of the square root, one of which is positive and denoted
$$
\sqrt {x_0},
$$
and another which is negative and denoted
$$
-\sqrt {x_0}.
$$
These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single smooth curve. It is therefore often useful to consider these two square root functions as a single function that has two values for positive , one value for 0 and no value for negative .
In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general.
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In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the implicit function that maps to a root of
$$
x^3-3x-y =0
$$
(see the figure on the right). For one may choose either
$$
0, \sqrt 3,\text{ or } -\sqrt 3
$$
for . By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval and the image is ; for the second one, the domain is and the image is ; for the last one, the domain is and the image is . As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single multi-valued function of that has three values for , and only one value for and .
Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically analytic functions. The domain to which a complex function may be extended by analytic continuation generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values.
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The domain to which a complex function may be extended by analytic continuation generally consists of almost the whole complex plane. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets . There are generally two ways of solving the problem. One may define a function that is not continuous along some curve, called a branch cut. Such a function is called the principal value of the function. The other way is to consider that one has a multi-valued function, which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the monodromy.
## In the foundations of mathematics
The definition of a function that is given in this article requires the concept of set, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.
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This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions.
For example, the singleton set may be considered as a function
$$
x\mapsto \{x\}.
$$
Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definitions for these weakly specified functions.
These generalized functions may be critical in the development of a formalization of the foundations of mathematics. For example, Von Neumann–Bernays–Gödel set theory, is an extension of the set theory in which the collection of all sets is a class. This theory includes the replacement axiom, which may be stated as: If is a set and is a function, then is a set.
In alternative formulations of the foundations of mathematics using type theory rather than set theory, functions are taken as primitive notions rather than defined from other kinds of object.
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This theory includes the replacement axiom, which may be stated as: If is a set and is a function, then is a set.
In alternative formulations of the foundations of mathematics using type theory rather than set theory, functions are taken as primitive notions rather than defined from other kinds of object. They are the inhabitants of function types, and may be constructed using expressions in the lambda calculus.
## In computer science
In computer programming, a function is, in general, a subroutine which implements the abstract concept of function. That is, it is a program unit that produces an output for each input. Functional programming is the programming paradigm consisting of building programs by using only subroutines that behave like mathematical functions, meaning that they have no side effects and depend only on their arguments: they are referentially transparent. For example, `if_then_else` is a function that takes three (nullary) functions as arguments, and, depending on the value of the first argument (true or false), returns the value of either the second or the third argument. An important advantage of functional programming is that it makes easier program proofs, as being based on a well founded theory, the lambda calculus (see below). However, side effects are generally necessary for practical programs, ones that perform input/output.
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An important advantage of functional programming is that it makes easier program proofs, as being based on a well founded theory, the lambda calculus (see below). However, side effects are generally necessary for practical programs, ones that perform input/output. There is a class of purely functional languages, such as Haskell, which encapsulate the possibility of side effects in the type of a function. Others, such as the ML family, simply allow side effects.
In many programming languages, every subroutine is called a function, even when there is no output but only side effects, and when the functionality consists simply of modifying some data in the computer memory.
Outside the context of programming languages, "function" has the usual mathematical meaning in computer science. In this area, a property of major interest is the computability of a function. For giving a precise meaning to this concept, and to the related concept of algorithm, several models of computation have been introduced, the old ones being general recursive functions, lambda calculus, and Turing machine. The fundamental theorem of computability theory is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The Church–Turing thesis is the claim that every philosophically acceptable definition of a computable function defines also the same functions.
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The fundamental theorem of computability theory is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The Church–Turing thesis is the claim that every philosophically acceptable definition of a computable function defines also the same functions.
General recursive functions are partial functions from integers to integers that can be defined from
- constant functions,
- successor, and
- projection functions
via the operators
- composition,
- primitive recursion, and
- minimization.
Although defined only for functions from integers to integers, they can model any computable function as a consequence of the following properties:
- a computation is the manipulation of finite sequences of symbols (digits of numbers, formulas, etc.),
- every sequence of symbols may be coded as a sequence of bits,
- a bit sequence can be interpreted as the binary representation of an integer.
Lambda calculus is a theory that defines computable functions without using set theory, and is the theoretical background of functional programming. It consists of terms that are either variables, function definitions (-terms), or applications of functions to terms.
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Lambda calculus is a theory that defines computable functions without using set theory, and is the theoretical background of functional programming. It consists of terms that are either variables, function definitions (-terms), or applications of functions to terms. Terms are manipulated by interpreting its axioms (the -equivalence, the -reduction, and the -conversion) as rewriting rules, which can be used for computation.
In its original form, lambda calculus does not include the concepts of domain and codomain of a function. Roughly speaking, they have been introduced in the theory under the name of type in typed lambda calculus. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus.
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Data cleansing or data cleaning is the process of identifying and correcting (or removing) corrupt, inaccurate, or irrelevant records from a dataset, table, or database. It involves detecting incomplete, incorrect, or inaccurate parts of the data and then replacing, modifying, or deleting the affected data. Data cleansing can be performed interactively using data wrangling tools, or through batch processing often via scripts or a data quality firewall.
After cleansing, a data set should be consistent with other similar data sets in the system. The inconsistencies detected or removed may have been originally caused by user entry errors, by corruption in transmission or storage, or by different data dictionary definitions of similar entities in different stores. Data cleaning differs from data validation in that validation almost invariably means data is rejected from the system at entry and is performed at the time of entry, rather than on batches of data.
The actual process of data cleansing may involve removing typographical errors or validating and correcting values against a known list of entities. The validation may be strict (such as rejecting any address that does not have a valid postal code), or with fuzzy or approximate string matching (such as correcting records that partially match existing, known records). Some data cleansing solutions will clean data by cross-checking with a validated data set.
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The validation may be strict (such as rejecting any address that does not have a valid postal code), or with fuzzy or approximate string matching (such as correcting records that partially match existing, known records). Some data cleansing solutions will clean data by cross-checking with a validated data set. A common data cleansing practice is data enhancement, where data is made more complete by adding related information. For example, appending addresses with any phone numbers related to that address. Data cleansing may also involve harmonization (or normalization) of data, which is the process of bringing together data of "varying file formats, naming conventions, and columns", and transforming it into one cohesive data set; a simple example is the expansion of abbreviations ("st, rd, etc." to "street, road, etcetera").
## Motivation
Administratively incorrect, inconsistent data can lead to false conclusions and misdirect investments on both public and private scales. For instance, the government may want to analyze population census figures to decide which regions require further spending and investment on infrastructure and services. In this case, it will be important to have access to reliable data to avoid erroneous fiscal decisions. In the business world, incorrect data can be costly. Many companies use customer information databases that record data like contact information, addresses, and preferences.
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