Proof by Contradiction Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Proof by Contradiction.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Proof by contradiction assumes the negation of your claim, follows valid logic until it produces something impossibly false, and concludes the original claim must be true.

Common stuck point: The procedure for proof by contradiction is the easy part; the trap is negating only part of the claim. Asking "Did I assume the claim is FALSE and then derive a genuinely impossible statement from it?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Did I assume the claim is FALSE and then derive a genuinely impossible statement from it?

Worked Examples

Example 1

medium

Prove by contradiction that

\sqrt{2}

is irrational.

Answer

\sqrt{2} \text{ is irrational.}

First step

Assume for contradiction that

\sqrt{2}

is rational. Then

\sqrt{2} = \frac{a}{b}

where

a, b

are integers with

\gcd(a,b)=1

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Example 2

medium

Prove by contradiction that there are infinitely many prime numbers.

Example 3

medium

Prove by contradiction that

\sqrt 3

is irrational.

Example 4

medium

Prove by contradiction that if

n^3

is even then

n

is even.

Example 5

medium

Prove by contradiction:

\sqrt 2 + 1

is irrational.

Example 6

medium

Prove by contradiction:

\sqrt 6

is irrational.

Example 7

medium

Prove by contradiction that

\log_{10} 2

is irrational.

Example 8

hard

Prove by contradiction: if

x

is irrational and

r\ne 0

is rational, then

rx

is irrational.

Example 9

hard

Prove by contradiction:

\sqrt 2 + \sqrt 5

is irrational.

Example 10

hard

Prove by contradiction: the set of real numbers in

[0,1]

is uncountable (Cantor's diagonal).

Example 11

challenge

Prove by contradiction:

\sqrt 2

cannot be written as

a + b\sqrt 3

with

a,b

rational and

b\ne 0

Example 12

challenge

Prove by contradiction: in any finite simple graph with

n\ge 2

vertices, at least two vertices have the same degree.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

hard

Prove by contradiction: There is no largest integer.

Example 2

medium

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

Example 3

easy

In proof by contradiction, what do you assume at the start?

Example 4

easy

To prove 'there is no largest integer' by contradiction, what is the assumption?

Example 5

easy

After assuming

\neg P

you derive '

0=1

'. What can you conclude?

Example 6

easy

Which assumption is correct to prove '

\sqrt2

is irrational' by contradiction?

Example 7

easy

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

Example 8

easy

How does proof by contradiction differ from proof by contrapositive?

Example 9

easy

True or false: a contradiction is reached when you derive both

R

and

\neg R

Example 10

easy

To prove 'if

n^2

is odd then

n

is odd' by contradiction, assume what?

Example 11

medium

Prove

\sqrt2

is irrational: complete the parity argument from

\sqrt2=a/b

(lowest terms).

Example 12

medium

Prove by contradiction: there is no smallest positive rational number.

Example 13

medium

Prove by contradiction: if

a+b \ge 100

then

a\ge 50

b\ge 50

Example 14

medium

Prove by contradiction:

\log_2 3

is irrational.

Example 15

medium

Identify the error: a student 'proves'

P

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

Why must you state the assumption explicitly at the start of a contradiction proof?

Example 18

medium

Prove by contradiction: if

n^2

is even then

n

is even.

Example 19

medium

Prove by contradiction: a triangle cannot have two right angles.

Example 20

challenge

Prove by contradiction that

\sqrt2 + \sqrt3

is irrational.

Example 21

challenge

Prove by contradiction: if

n

pigeons occupy

n-1

holes, some hole has

\ge 2

pigeons.

Example 22

challenge

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

Example 23

easy

To prove by contradiction that 'no integer is both even and odd,' what assumption do you start with?

Example 24

easy

Negate (for a contradiction proof): 'There is no integer between

0

and

1

Example 25

easy

For 'there is no rational number

r

with

r^2 = 5

', what is the contradiction-assumption?

Example 26

medium

Prove by contradiction: if

a,b

are positive reals with

a+b<2

, then

ab<1

Example 27

medium

Prove by contradiction: there is no greatest negative real number.

Example 28

medium

Prove by contradiction: if

a,b

are integers with

a^2 + b^2 = 4k+3

for some integer

k

, this is impossible.

Example 29

medium

Prove by contradiction: among any

3

consecutive integers, at least one is divisible by

3

Example 30

hard

Prove by contradiction: there are infinitely many primes of the form

4k+3

Example 31

hard

Prove by contradiction: if

7

pigeons are placed in

3

holes, some hole contains at least

3

pigeons.

Example 32

hard

Prove by contradiction: in any group of

6

people, either

3

mutually know each other or

3

are mutual strangers.

Example 33

hard

Prove by contradiction: no rational number satisfies

x^3 = 2

Example 34

challenge

Prove by contradiction:

2^n > n^2

for all integers

n\ge 5

, fails if we drop the hypothesis

n\ge 5

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Related Concepts

Contradiction Logical Statement Direct Proof

Background Knowledge

These ideas may be useful before you work through the harder examples.

contradictionlogical statementdirect proof