Practice question · Multiple choice
"If n is divisible by 4 then n is even" is true, but its converse is false. Why does a true implication give no guarantee at all about its converse, while its contrapositive is automatically true?
Hints
- An implication fails only when the premise is true and the conclusion false. Check when each of the three fails.
- n = 6 is even and not divisible by 4. Which of the three statements does it refute?
Show the answer
C. Because p → q and ¬q → ¬p fail in exactly the same case.
Why
Compare the failure conditions: p → q fails when p is true and q false, and ¬q → ¬p fails in precisely that same case, so the two stand or fall together. The converse fails in a different situation, and n = 6 supplies it. Converses are sometimes true, and when both hold you have an equivalence, which is what a characterisation theorem asserts.
Practise Implication and Equivalence
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