Proof by contrapositive
Master proof by contrapositive for Year 11 Specialist Mathematics in Queensland (QCAA). Rather than proving a statement directly, you prove its equivalent contrapositive, which swaps and negates the original.
You will learn to build the converse, inverse and contrapositive of a statement, see why only the contrapositive is equivalent to the original, and use it to prove results about integers and inequalities.
Theory
Proof by contrapositive proves a statement of the form "if \(P\) then \(Q\)" by instead proving the logically equivalent contrapositive "if not \(Q\) then not \(P\)" in Year 11 Specialist Mathematics (QCAA, Queensland). This page shows how the contrapositive is built, how it differs from the converse and inverse, and how to use it in a proof.
A conditional statement has the form \(P\Rightarrow Q\), read "if \(P\), then \(Q\)". Here \(P\) is the hypothesis and \(Q\) is the conclusion. From it we can build three related statements.
The converse is \(Q\Rightarrow P\): you swap the hypothesis and conclusion. The inverse is \(\neg P\Rightarrow\neg Q\): you negate both parts but keep the order. The contrapositive is \(\neg Q\Rightarrow\neg P\): you negate both parts and swap them.
The key fact is that a statement is logically equivalent to its contrapositive: \(P\Rightarrow Q\) and \(\neg Q\Rightarrow\neg P\) are always true or false together. The converse and inverse are equivalent to each other, but neither is equivalent to the original.
To prove by contrapositive, assume \(\neg Q\) and reason directly until you reach \(\neg P\). Because the contrapositive is equivalent to the original, this proves \(P\Rightarrow Q\). This is useful when \(\neg Q\) is an easier place to start than \(P\).
Starting from the conditional \(P\Rightarrow Q\), the related forms are:
The equivalence that makes the method work is:
How to prove a statement by contrapositive
- Identify \(P\) (the hypothesis) and \(Q\) (the conclusion) in the statement "if \(P\), then \(Q\)".
- Write the contrapositive \(\neg Q\Rightarrow\neg P\): negate both parts and swap them (do not just swap — that gives the converse).
- Assume \(\neg Q\) and reason directly, step by step, using algebra or definitions.
- Reach \(\neg P\). The contrapositive is proved, so the original statement is proved too.
Identify the parts, negate each, then swap:
| \(P\) | \(:\) | \(\text{it is snowing}\) |
| \(Q\) | \(:\) | \(\text{it is cold}\) |
| \(\neg Q\) | \(:\) | \(\text{it is not cold}\) |
| \(\neg P\) | \(:\) | \(\text{it is not snowing}\) |
| \(\text{contrapositive}\) | \(:\) | \(\neg Q \Rightarrow \neg P\) |
The contrapositive is: "If it is not cold, then it is not snowing."
The converse swaps only; the contrapositive negates both and swaps:
| \(\text{original}\) | \(:\) | \(P \Rightarrow Q\) |
| \(\text{converse}\) | \(:\) | \(Q \Rightarrow P \ \ (\text{swap})\) |
| \(\text{contrapositive}\) | \(:\) | \(\neg Q \Rightarrow \neg P \ \ (\text{negate and swap})\) |
Converse: "If \(n\) is a multiple of \(3\), then \(n\) is a multiple of \(9\)" — this is false (take \(n=6\)). Contrapositive: "If \(n\) is not a multiple of \(3\), then \(n\) is not a multiple of \(9\)" — true, like the original.
Converse (false): multiple of 3 ⇒ multiple of 9. Contrapositive (true): not a multiple of 3 ⇒ not a multiple of 9.
State the contrapositive (negate both and swap):
"If \(n\) is even, then \(3n+5\) is odd." Assume \(n=2k\):
| \(3n+5\) | \(=\) | \(3(2k)+5\) |
| \(=\) | \(6k+5\) | |
| \(=\) | \(2(3k+2)+1\) |
This has the form \(2m+1\), so \(3n+5\) is odd. The contrapositive holds.
The contrapositive is proved, so the original statement is true.
Test the converse "if \(x^2>25\) then \(x>5\)" with \(x=-6\):
| \(x\) | \(=\) | \(-6\) |
| \(x^2\) | \(=\) | \(36\) |
| \(36\) | \(>\) | \(25 \ \ (\text{hypothesis holds})\) |
| \(x=-6\) | \(\not>\) | \(5 \ \ (\text{conclusion fails})\) |
So the converse is false. The contrapositive "if \(x^2\le 25\), then \(x\le 5\)" is true, matching the original.
The converse is false; the contrapositive is true and equivalent to the original.
Common pitfalls
Frequently asked questions
What is the contrapositive of a statement?
The contrapositive of "if \(P\) then \(Q\)" is "if not \(Q\) then not \(P\)": you negate both parts and swap them. It is logically equivalent to the original statement.
What is the difference between the converse, inverse and contrapositive?
The converse \(Q\Rightarrow P\) swaps the parts; the inverse \(\neg P\Rightarrow\neg Q\) negates both parts; the contrapositive \(\neg Q\Rightarrow\neg P\) negates both parts and swaps. Only the contrapositive is equivalent to the original.
Why does proof by contrapositive work?
Because a conditional and its contrapositive always have the same truth value. If you prove \(\neg Q\Rightarrow\neg P\), you have automatically proved the equivalent statement \(P\Rightarrow Q\).
Is the contrapositive the same as the converse?
No. The converse only swaps the hypothesis and conclusion; the contrapositive swaps them and negates both. The converse is not equivalent to the original, but the contrapositive is.
How do you start a proof by contrapositive?
Assume the negation of the conclusion, \(\neg Q\), then reason directly with algebra or definitions until you reach the negation of the hypothesis, \(\neg P\).
How do you negate an inequality like \(x>3\)?
The negation of \(x>3\) is \(x\le 3\). The negation includes the boundary, so it is "less than or equal to", not just "less than".