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    "### Data Analysis\n",
    "#### Inclusion-exclusion principle \n",
    "<hr>\n",
    "https://github.com/ostad-ai/Machine-Learning\n",
    "<br> Explanation: https://www.pinterest.com/HamedShahHosseini/Machine-Learning/background-knowledge"
   ]
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  {
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   "source": [
    "For set operaions in Python, see our repository for Python\n",
    "<br> https://github.com/ostad-ai/Python-Everything"
   ]
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    "In set theory: \n",
    "- $|A \\cup B|=|A|+|B|-|A \\cap B|$\n",
    "<hr>\n",
    "\n",
    "In probability:\n",
    "- $P(A\\cup B)=P(A)+P(B)-P(A \\cap B)$"
   ]
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     "text": [
      "Checking inclusion-exclusion principle for sets:\n",
      "It is True\n"
     ]
    }
   ],
   "source": [
    "# for a given set A\n",
    "def cardinality(A):\n",
    "    return len(A)\n",
    "#-----for sets A and B\n",
    "def union(A,B):\n",
    "    return A|B\n",
    "def intersection(A,B):\n",
    "    return A&B\n",
    "#---example:\n",
    "A={2,4,6} # dice with even outcomes\n",
    "B={1,2,3}  # dice less than 4\n",
    "S={1,2,3,4,5,6} #sampel space\n",
    "print('Checking inclusion-exclusion principle for sets:')\n",
    "result=cardinality(union(A,B))==cardinality(A)+cardinality(B)-cardinality(intersection(A,B))\n",
    "print(f'It is {result}')"
   ]
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   "execution_count": 2,
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     "text": [
      "Checking inclusion-exclusion principle for probabilities:\n",
      "It is True\n"
     ]
    }
   ],
   "source": [
    "#-----probability of event A in sample space S\n",
    "def prob(A,S):\n",
    "    return cardinality(A)/cardinality(S)\n",
    "#--- example: \n",
    "print('Checking inclusion-exclusion principle for probabilities:')\n",
    "result= prob(union(A,B),S)==prob(A,S)+prob(B,S)-prob(intersection(A,B),S)\n",
    "print(f'It is {result}')"
   ]
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