Contrapositive Formula
The Formula
(P \to Q) \Leftrightarrow (\neg Q \to \neg P)
When to use: Flip and negate. Always has the same truth value as the original.
Quick Example
Original: 'If it rains, the ground is wet.' Contrapositive: 'If the ground isn't wet, it didn't rain.'
Notation
\sim Q \to \sim P is the contrapositive of P \to Q
What This Formula Means
The statement 'If not Q, then not P'—logically equivalent to 'If P, then Q.'
Flip and negate. Always has the same truth value as the original.
Formal View
(P \to Q) \Leftrightarrow (\neg Q \to \neg P); both have identical truth tables in all four rows
Worked Examples
Example 1
easyForm the contrapositive of: 'If n^2 is even, then n is even.'
Solution
- 1 Recall the structure of the conditional: p \Rightarrow q where p: 'n^2 is even' and q: 'n is even.'
- 2 The contrapositive is \neg q \Rightarrow \neg p. Form the negations: \neg q: 'n is not even' (i.e., n is odd), and \neg p: 'n^2 is not even' (i.e., n^2 is odd).
- 3 Contrapositive: 'If n is odd, then n^2 is odd.' By the logical equivalence p \Rightarrow q \equiv \neg q \Rightarrow \neg p, this statement has exactly the same truth value as the original — and it is in fact true (odd times odd is odd).
Answer
\text{If } n \text{ is odd, then } n^2 \text{ is odd.}
The contrapositive negates both parts and swaps them. It is always logically equivalent to the original conditional, making it a powerful tool in proofs.
Example 2
mediumProve by contrapositive: 'If n^2 is odd, then n is odd.'
Common Mistakes
- Mixing up contrapositive (\neg Q \to \neg P) with converse (Q \to P) — only the contrapositive is logically equivalent
- Negating only the hypothesis or only the conclusion instead of both — the contrapositive flips AND negates both parts
- Thinking the inverse (\neg P \to \neg Q) is the same as the contrapositive — the inverse is equivalent to the converse, not the original
Why This Formula Matters
Often easier to prove than the original; valid proof technique.
Frequently Asked Questions
What is the Contrapositive formula?
The statement 'If not Q, then not P'—logically equivalent to 'If P, then Q.'
How do you use the Contrapositive formula?
Flip and negate. Always has the same truth value as the original.
What do the symbols mean in the Contrapositive formula?
\sim Q \to \sim P is the contrapositive of P \to Q
Why is the Contrapositive formula important in Math?
Often easier to prove than the original; valid proof technique.
What do students get wrong about Contrapositive?
Contrapositive \neq converse. Converse: 'If Q, then P'—NOT equivalent.
What should I learn before the Contrapositive formula?
Before studying the Contrapositive formula, you should understand: conditional, negation.