If P → Q is true, what can be said about the contrapositive ¬Q → ¬P?

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Multiple Choice

If P → Q is true, what can be said about the contrapositive ¬Q → ¬P?

The key idea is that a conditional and its contrapositive always share the same truth value. If P implies Q is true, then not Q implies not P must also be true. This happens because a conditional is only false when P is true and Q is false; in that same situation, the contrapositive would be not Q (true) implies not P (false), which is also false. Since that problematic case never occurs when the original is true, every time P → Q holds, the contrapositive ¬Q → ¬P holds as well. So the contrapositive is also true. For example, if “If it is raining, the ground is wet” is true, then “If the ground is not wet, it is not raining” is true too.

The key idea is that a conditional and its contrapositive always share the same truth value. If P implies Q is true, then not Q implies not P must also be true. This happens because a conditional is only false when P is true and Q is false; in that same situation, the contrapositive would be not Q (true) implies not P (false), which is also false. Since that problematic case never occurs when the original is true, every time P → Q holds, the contrapositive ¬Q → ¬P holds as well. So the contrapositive is also true. For example, if “If it is raining, the ground is wet” is true, then “If the ground is not wet, it is not raining” is true too.

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