Contradiction Examples in Math

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

A mathematical statement that is always false — no values of the variables can ever make it true.

$x + y = 5$ AND $x + y = 7$ can't both be true simultaneously — this is a contradiction.

Read the full concept explanation →

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: A contradiction is a claim no values can ever satisfy, like $0=3$ .

Common stuck point: The procedure for contradiction is the easy part; the trap is reading $0=0$ as a contradiction. Asking "Have I reached a statement that no values could ever make true?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Have I reached a statement that no values could ever make true?

Worked Examples

Example 1

easy

Show that

x + 1 = x + 3

is a contradiction.

Answer

Contradiction — no solution.

First step

Step 1: Subtract

x

from both sides:

1 = 3

Full solution

2
Step 2: $1 \neq 3$ . This is always false.
3
Step 3: No value of $x$ can make this true — it's a contradiction.

A contradiction arises when algebraic manipulation leads to a false statement like

0 = c

where

c \neq 0

. It means the original equation (or system) has no solution.

Example 2

medium

Solve

2(x + 3) = 2x + 5

Example 3

medium

Solve

\dfrac{2}{x-1} = \dfrac{2}{x-1} + 5

Example 4

medium

Solve

5(x-2) - 3x = 2x + 7

Example 5

hard

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

Example 6

hard

Prove by contradiction that

\sqrt{2}

is irrational.

Example 7

challenge

Prove by contradiction that there are infinitely many primes.

Practice Problems

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

Example 1

easy

0 = 7

a contradiction or an identity?

Example 2

medium

Solve

|x| = -5

Example 3

easy

0 = 5

ever true?

Example 4

easy

0 = 0

a contradiction or an identity?

Example 5

easy

Can

x + 3 = x + 7

have a solution?

Example 6

easy

System reduces to

0 = -2

. How many solutions?

Example 7

easy

Two parallel lines give a system whose elimination yields what kind of statement?

Example 8

easy

Is 'this number is both even and odd' a contradiction?

Example 9

easy

Does reaching

0 = 3

mean you made an arithmetic error?

Example 10

easy

Which has NO solution:

\{x+y=2,\ x+y=2\}

\{x+y=2,\ x+y=5\}

Example 11

medium

Solve

\{2x + y = 4,\ 4x + 2y = 9\}

and interpret.

Example 12

medium

After elimination you get

0 = 0

. Is this a contradiction?

Example 13

medium

For what

b

does

\{x + y = 3,\ x + y = b\}

contain a contradiction?

Example 14

medium

|x| = -4

a contradiction?

Example 15

medium

A proof assumes

\sqrt{2}=\tfrac{p}{q}

in lowest terms and derives that

p,q

are both even. Why is this a contradiction?

Example 16

medium

Does

\{x > 5,\ x < 2\}

have a solution?

Example 17

medium

Elimination on a

3\times3

system yields

0 = 1

in one row. What can you conclude?

Example 18

challenge

Show

\{x + 2y = 1,\ 3x + 6y = 5\}

is contradictory.

Example 19

challenge

Find all

a

for which

\{x + y = 1,\ ax + ay = 3\}

is contradictory.

Example 20

challenge

Explain why a contradiction makes EVERY conclusion derivable in classical logic.

Example 21

medium

Does

\{2x = 6,\ 2x = 10\}

contain a contradiction?

Example 22

medium

Reaching

5 = 8

while solving an equation means what?

Example 23

easy

Solve

x + 5 = x + 2

Example 24

easy

How many solutions does

3(x+2) = 3x + 6

have?

Example 25

easy

Does

x^2 = -9

have a real solution?

Example 26

easy

Does the system

\{x = 4,\ x = 7\}

have a solution?

Example 27

medium

Solve the system

\{3x - y = 4,\ 6x - 2y = 5\}

Example 28

medium

For what value of

k

is the system

\{2x + 3y = 6,\ 4x + 6y = k\}

contradictory?

Example 29

medium

Solve

|x - 3| = -1

Example 30

medium

Does the system

\{x + y > 5,\ x + y < 2\}

have a solution?

Example 31

medium

Determine whether

\{x + 2y = 5,\ 2x + 4y = 9\}

has a solution.

Example 32

hard

Find all real

a

for which

\{x + y = 2,\ ax + ay = 6\}

has no solution.

Example 33

hard

Does

\sin x = 2

have a real solution?

Example 34

hard

Does

e^x = 0

have any real solution?

Example 35

medium

Solve

\dfrac{x+1}{x-1} = 1

for

x \ne 1

Example 36

medium

Does the system

\{y = 2x + 1,\ y = 2x - 3\}

have a solution?

Example 37

challenge

Show

\{x^2 + y^2 = 1,\ x^2 + y^2 = 4\}

has no solution.

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Related Concepts

Equations Consistency

Background Knowledge

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

equations