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1.1 no 41
😃: State the converse, contrapositive, and inverse of each of
these conditional statements.
a) If it snows today, I will ski tomorrow.
b) I come to class whenever there is going to be a quiz.
c) A positive integer is a prime only if it has no divisors
other than 1 and itself.
🤖: The given statement can be analyzed as follows:
(p ∨ q ∨ r) ∧ (¬p ∨ ¬q ∨ ¬r)
= (p OR q OR r) AND (NOT p OR NOT q OR NOT r)
When at least one of p, q, and r is true and at least one is false, we can break down the two parts of the conjunction separately:
- The first part (p OR q OR r): Since at least one of p, q, and r is true, this part is definitely true.
- The second part (NOT p OR NOT q OR NOT r): Since at least one of p, q, and r is false, this part is also definiteley true.
Therefore, the entire expression is true when at least one of p, q, and r is true and at least one is false.
Now, let's consider what happens when all three variables have the same truth value. In that case, both parts of the conjunction would evaluate to the same boolean value, which means their logical AND operation would result in FALSE.
So, the statement is true when at least one of p, q, and r is true and at least one is false, and false when all three variables have the same truth value.