Proof by Contrapositive
Prove ¬q → ¬p instead of p → q.
Key idea
Prove ¬q → ¬p instead of p → q.
In depth
To prove p → q, prove the equivalent ¬q → ¬p. Often easier when the negation of the conclusion gives something concrete to work with.
Worked example
If n² is even then n is even. Contrapositive: if n is odd, n = 2k + 1, n² = 4k² + 4k + 1 is odd.
Real-world uses
- Proving a number is not prime when its square is not prime in maths coursework
Common mistake
Proving the converse instead.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Proof by Contrapositive?
Prove ¬q → ¬p instead of p → q. It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Proof by Contrapositive used in real life?
Real-world examples: Proving a number is not prime when its square is not prime in maths coursework.
What should I learn before Proof by Contrapositive?
Make sure you are comfortable with Proof by Direct Argument, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Proof by Contrapositive?
Proving the converse instead.
How can I practise Proof by Contrapositive?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Proof by Contrapositive?
Move on to Proof by Contradiction next.
Related advanced topics in Discrete Math & Number Theory
- Tautologies and Contradictions – Always true or always false statements.
- Logical Equivalence – Statements with identical truth tables.
- De Morgan’s Laws – Negation swaps AND and OR.
- Converse and Contrapositive – A conditional equals its contrapositive.
- Quantifiers – “For all” and “there exists”.
- Negating Quantified Statements – Flip the quantifier and negate the predicate.