Practice Proof by Contradiction in Math

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

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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.

Showing a random 20 of 50 problems.

Example 1

challenge
Prove by contradiction that 2+3\sqrt2 + \sqrt3 is irrational.

Example 2

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

Example 3

medium
Prove by contradiction: log⁡23\log_2 3 is irrational.

Example 4

medium
Prove by contradiction: among any 33 consecutive integers, at least one is divisible by 33.

Example 5

easy
For 'there is no rational number rr with r2=5r^2 = 5', what is the contradiction-assumption?

Example 6

medium
Prove by contradiction: 6\sqrt 6 is irrational.

Example 7

easy
Negate the statement to be assumed for contradiction: 'Every even number greater than 2 is composite.'

Example 8

challenge
Prove by contradiction: 2n>n22^n > n^2 for all integers n≥5n\ge 5, fails if we drop the hypothesis n≥5n\ge 5.

Example 9

hard
Prove by contradiction: in any group of 66 people, either 33 mutually know each other or 33 are mutual strangers.

Example 10

hard
Prove by contradiction: if 77 pigeons are placed in 33 holes, some hole contains at least 33 pigeons.

Example 11

hard
Prove by contradiction: there are infinitely many primes of the form 4k+34k+3.

Example 12

challenge
Prove by contradiction: there is no rational number whose square is 3.

Example 13

easy
Which assumption is correct to prove '2\sqrt2 is irrational' by contradiction?

Example 14

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

Example 15

medium
Identify the error: a student 'proves' PP by contradiction but the derived statements never actually conflict.

Example 16

medium
Prove by contradiction: between any two distinct reals there is another real.

Example 17

medium
Prove by contradiction: if a,ba,b are positive reals with a+b<2a+b<2, then ab<1ab<1.

Example 18

easy
Negate (for a contradiction proof): 'There is no integer between 00 and 11.'

Example 19

hard
Prove by contradiction: no rational number satisfies x3=2x^3 = 2.

Example 20

easy
To prove 'if n2n^2 is odd then nn is odd' by contradiction, assume what?
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