Proof by Contradiction
Assume the opposite and reach an impossibility.
Key idea
Assume the opposite and reach an impossibility.
In depth
Assume the statement is false and derive an impossibility. Since logic cannot produce a contradiction from a true premise, the assumption must be wrong. Classic examples: √2 irrational and infinitely many primes.
Worked example
Suppose √2 = p/q in lowest terms. Then p² = 2q², so p is even, then q is even too, contradicting lowest terms.
Real-world uses
- Proving that √2 is irrational and that there are infinitely many primes
Common mistake
Not clearly identifying the contradiction reached.
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 Contradiction?
Assume the opposite and reach an impossibility. It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Proof by Contradiction used in real life?
Real-world examples: Proving that √2 is irrational and that there are infinitely many primes.
What should I learn before Proof by Contradiction?
Make sure you are comfortable with Proof by Contrapositive, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Proof by Contradiction?
Not clearly identifying the contradiction reached.
How can I practise Proof by Contradiction?
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 Contradiction?
Move on to Proof by Cases 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.