Practice Proof by Contradiction in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

Proof by contradiction (reductio ad absurdum) assumes the negation of what you want to prove, then derives a logical contradiction, thereby establishing that the original statement must be true.

Assume the opposite of what you want to prove, then follow the logic to a statement that is impossibly false — proving your assumption must have been wrong.

Example 1

medium
Prove by contradiction that \sqrt{2} is irrational.

Example 2

medium
Prove by contradiction that there are infinitely many prime numbers.

Example 3

hard
Prove by contradiction: There is no largest integer.

Example 4

medium
Prove by contradiction that there is no smallest positive rational number.
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