Negation Examples in Math

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

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

Concept Recap

The negation of a statement $P$ , written $\neg P$ , is the statement with the opposite truth value: true when $P$ is false, and false when $P$ is true.

Flipping true to false or false to true. 'It is NOT the case that...'

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: The negation of P is the statement that is true exactly when P is false.

Common stuck point: The procedure for negation is the easy part; the trap is negating 'all are' as 'none are'. Asking "Is this new statement true in exactly the cases where the original is false?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this new statement true in exactly the cases where the original is false?

Worked Examples

Example 1

easy

Write the negation of each statement and determine its truth value: (a) '

5 > 3

', (b) 'All cats are black.'

Answer

(a)\;5 \le 3 \;(\text{False}),\quad (b)\;\text{Some cat is not black}\;(\text{True})

First step

Recall that the negation

\neg P

of a statement

P

is the statement that is true exactly when

P

is false.

Full solution

2
(a) $P$ : ' $5 > 3$ ' (True). The negation reverses the inequality: $\neg P$ : ' $5 \le 3$ ' (False). (b) $P$ : 'All cats are black' has form $\forall x, P(x)$ . Its negation is $\exists x, \neg P(x)$ : 'There exists a cat that is not black.'
3
Truth values: (a) $\neg P$ is False because $5 > 3$ is true. (b) $\neg P$ is True because black cats are not the only kind — there exist non-black cats in the world.

Negation flips the truth value. For universal statements (

\forall

), the negation is an existential statement (

\exists

). The original and its negation always have opposite truth values.

Example 2

medium

Simplify

\neg(\neg p \lor q)

using logical laws.

Example 3

medium

Use De Morgan's law to negate: '

x>2

AND

x<10

Example 4

medium

Show that

\neg(\neg(\neg P))

simplifies to

\neg P

Example 5

medium

Find the negation of '

P \Leftrightarrow Q

' (biconditional).

Example 6

hard

Write the contrapositive of 'If

n^2

is odd, then

n

is odd', then explain how it relates to the original by negation.

Example 7

hard

Show that

\neg(P \Rightarrow Q) \Rightarrow P

is a tautology.

Example 8

challenge

Prove by negation: show that '

\sqrt{2}

is irrational' is logically equivalent to the negation of 'there exist integers

p, q

with

q \neq 0

and

p/q = \sqrt{2}

in lowest terms.'

Practice Problems

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

Example 1

easy

Write the negation of: (a) 'It is raining', (b) '

x \ge 5

', (c) 'Every student passed.'

Example 2

medium

Build the truth table for

\neg p

and verify that

p \land \neg p

is always false (contradiction) and

p \lor \neg p

is always true (tautology).

Example 3

easy

P

is true, what is the truth value of

\neg P

Example 4

easy

Negate: 'It is raining.'

Example 5

easy

Simplify

\neg(\neg P)

Example 6

easy

Negate: 'All cats are black.'

Example 7

easy

Negate: 'Some students passed.'

Example 8

easy

What is the truth value of

\neg P

when

P

is false?

Example 9

easy

Negate the inequality

x > 5

Example 10

easy

Negate: 'The number is even.'

Example 11

medium

Negate: 'All primes are odd.' Is the negation true?

Example 12

medium

Use De Morgan to negate

P \wedge Q

Example 13

medium

Use De Morgan to negate

P \vee Q

Example 14

medium

Negate: 'If it rains, then the game is cancelled.'

Example 15

medium

Negate: 'For every

x

, there exists

y

such that

y > x

Example 16

medium

Negate: 'The function is continuous AND differentiable.'

Example 17

medium

Is the negation of 'exactly one of

P,Q

is true' equal to '

P\Leftrightarrow Q

Example 18

medium

Negate: 'No student failed the exam.'

Example 19

medium

Negate: '

x \ge 3

AND

x \le 7

', then describe the result.

Example 20

challenge

Negate fully and simplify: 'For all

\varepsilon>0

, there exists

\delta>0

such that

|x-a|<\delta \Rightarrow |f(x)-L|<\varepsilon

Example 21

challenge

Negate: 'Every even integer greater than 2 is the sum of two primes' (Goldbach). State what a disproof would require.

Example 22

challenge

Show

\neg(P \Rightarrow Q)

is logically equivalent to

P \wedge \neg Q

using a truth table.

Example 23

easy

Write the negation of the statement: '

x = 4

Example 24

easy

Negate: 'The light is on.'

Example 25

easy

Negate the statement: '

y \leq 0

Example 26

medium

Negate: 'There is at least one black swan.'

Example 27

medium

Negate: 'Every triangle has three sides.'

Example 28

medium

Negate: '

x

is a multiple of 3 OR

x

is even.'

Example 29

medium

Write the negation of: 'If

n

is even, then

n^2

is even.'

Example 30

medium

Negate: 'At most 5 students passed.'

Example 31

medium

Negate: 'At least 3 cards are red.'

Example 32

medium

Build a truth table for

\neg(P \land Q)

and verify it agrees with

\neg P \lor \neg Q

Example 33

hard

Negate and simplify: '

\forall x \in \mathbb{R}, x^2 \geq 0

Example 34

hard

Negate: 'For all

n \in \mathbb{N}

, there exists a prime

p > n

Example 35

hard

Negate: '

x>0

AND (

y<0

z=5

).'

Example 36

hard

Negate: 'Every continuous function on

[0,1]

attains its maximum.'

Example 37

medium

Negate: 'Exactly one solution exists.'

Example 38

challenge

Negate and simplify: '

\forall \varepsilon > 0, \exists N \in \mathbb{N}, \forall n \geq N, |a_n - L| < \varepsilon

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Related Concepts

Logical Statement

Background Knowledge

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

logical statement