Contrapositive
The implication 'if not Q then not P'. It is true exactly when 'if P then Q' is true, so you can prove either one.
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Analogy
Reading a map backwards: if every road from A leads to B, then anyone not at B cannot have come from A.
Example
Original: 4 | n gives n even.Contrapositive: n odd gives 4 does not divide n.Converse: n even gives 4 | n.The converse fails for n = 2.
In practice
Use it when the 'not' version is easier to start from. Do not confuse it with the converse, which is a different claim.
Practice
Flashcards
- What is the contrapositive of 'if P then Q'?'If not Q then not P'. It always has the same truth value as the original.
- Is the converse of a true implication always true?No. 'If 4 divides n then n is even' is true, but its converse is false for n = 2.
Questions people ask
What is the difference between the contrapositive and the converse?
The contrapositive of 'if P then Q' is 'if not Q then not P' and always has the same truth value as the original. The converse is 'if Q then P' and can differ. 'If n is divisible by 4 then n is even' is true, but its converse is false.
When should I prove the contrapositive instead of the original?
Use it when the conclusion's negation gives you something concrete to work with. To show 'if n squared is even then n is even', assume n is odd, so n = 2k + 1. Then n squared is odd, and the original claim follows.