Wednesday, January 1, 2025

Is the contrapositive of "If P is true, then Q is true" "If Q is not true, then P is not true"?

 Yes, the contrapositive of "If 

PP is true, then QQ is true" is indeed "If QQ is not true, then PP is not true."

Here's why:

The statement "If PP, then QQ" can be written in logical form as:

P→QP \rightarrow Q

The contrapositive of a conditional statement P→QP \rightarrow Q is the statement obtained by negating both the hypothesis and the conclusion, and then reversing the direction of the implication. That is:

¬Q→¬P\neg Q \rightarrow \neg P

So, the contrapositive of "If PP is true, then QQ is true" (i.e., P→QP \rightarrow Q) is "If QQ is not true, then PP is not true" (i.e., ¬Q→¬P\neg Q \rightarrow \neg P).

In logic, the original statement and its contrapositive are logically equivalent, meaning they have the same truth value.

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