Contrapositive Formula

The contrapositive of a conditional statement P Q is Q P, formed by negating both parts and reversing their order — it is always logically equivalent to.

The Formula

(P→Q)⇔(¬Q→¬P)(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

∼Q→∼P\sim Q \to \sim P is the contrapositive of P→QP \to Q

What This Formula Means

The contrapositive of a conditional statement P⇒QP \Rightarrow Q is ¬Q⇒¬P\neg Q \Rightarrow \neg P, formed by negating both parts and reversing their order — it is always logically equivalent to the original.

Flip and negate. Always has the same truth value as the original.

Formal View

(P→Q)⇔(¬Q→¬P)(P \to Q) \Leftrightarrow (\neg Q \to \neg P); both have identical truth tables in all four rows

Worked Examples

Example 1

easy
Form the contrapositive of: 'If n2n^2 is even, then nn is even.'

Answer

If n is odd, then n2 is odd.\text{If } n \text{ is odd, then } n^2 \text{ is odd.}

First step

1
Recall the structure of the conditional: p⇒qp \Rightarrow q where pp: 'n2n^2 is even' and qq: 'nn is even.'

Full solution

  1. 2
    The contrapositive is ¬q⇒¬p\neg q \Rightarrow \neg p. Form the negations: ¬q\neg q: 'nn is not even' (i.e., nn is odd), and ¬p\neg p: 'n2n^2 is not even' (i.e., n2n^2 is odd).
  2. 3
    Contrapositive: 'If nn is odd, then n2n^2 is odd.' By the logical equivalence p⇒q≡¬q⇒¬pp \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).
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

medium
Prove by contrapositive: 'If n2n^2 is odd, then nn is odd.'

Example 3

medium
Prove by contrapositive: if 3n+23n + 2 is odd, then nn is odd.

Common Mistakes

  • Negating without reversing, giving the inverse ¬P→¬Q\neg P \to \neg Q — the contrapositive must reverse too.
  • Reversing without negating, giving the converse Q→PQ \to P — neither is equivalent to the original.
  • Forgetting the equivalence — the contrapositive always shares the original's truth value, so proving it proves the original.

Why This Formula Matters

The contrapositive is the workhorse of indirect proof: proving 'if not Q then not P' proves 'if P then Q' for free, because they are logically equivalent. A student who instead negates without reversing (the inverse) or only reverses (the converse) proves a non-equivalent statement and a broken proof. Recognizing it by "Did I negate BOTH parts AND reverse their order, keeping the same truth value?" — rather than by familiar numbers — is what lets a student tell it apart from converse and inverse and original conditional in a mixed problem set.

Frequently Asked Questions

What is the Contrapositive formula?

The contrapositive of a conditional statement P⇒QP \Rightarrow Q is ¬Q⇒¬P\neg Q \Rightarrow \neg P, formed by negating both parts and reversing their order — it is always logically equivalent to the original.

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?

∼Q→∼P\sim Q \to \sim P is the contrapositive of P→QP \to Q

Why is the Contrapositive formula important in Math?

The contrapositive is the workhorse of indirect proof: proving 'if not Q then not P' proves 'if P then Q' for free, because they are logically equivalent. A student who instead negates without reversing (the inverse) or only reverses (the converse) proves a non-equivalent statement and a broken proof. Recognizing it by "Did I negate BOTH parts AND reverse their order, keeping the same truth value?" — rather than by familiar numbers — is what lets a student tell it apart from converse and inverse and original conditional in a mixed problem set.

What do students get wrong about Contrapositive?

The procedure for contrapositive is the easy part; the trap is negating without reversing, giving the inverse ¬P→¬Q\neg P \to \neg Q. Asking "Did I negate BOTH parts AND reverse their order, keeping the same truth value?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Contrapositive formula?

Before studying the Contrapositive formula, you should understand: conditional, negation.

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