Practice Negation in Math

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

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Quick 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...'

Showing a random 20 of 50 problems.

Example 1

medium

Use De Morgan to negate

P \wedge Q

Example 2

easy

Negate the statement: '

y \leq 0

Example 3

medium

Is the negation of 'exactly one of

P,Q

is true' equal to '

P\Leftrightarrow Q

Example 4

medium

Negate: 'No student failed the exam.'

Example 5

easy

Write the negation of the statement: '

x = 4

Example 6

challenge

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

Example 7

easy

Simplify

\neg(\neg P)

Example 8

medium

Negate: 'Every triangle has three sides.'

Example 9

medium

Negate: 'At least 3 cards are red.'

Example 10

challenge

Negate and simplify: '

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

Example 11

easy

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

x \ge 5

', (c) 'Every student passed.'

Example 12

medium

Use De Morgan to negate

P \vee Q

Example 13

medium

Show that

\neg(\neg(\neg P))

simplifies to

\neg P

Example 14

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 15

medium

Negate: 'Exactly one solution exists.'

Example 16

medium

Use De Morgan's law to negate: '

x>2

AND

x<10

Example 17

medium

Negate: 'At most 5 students passed.'

Example 18

easy

Negate: 'It is raining.'

Example 19

medium

True or false: the negation of 'all dogs bark' is 'no dogs bark.'

Example 20

easy

Negate: 'The number is even.'

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Related Concepts

Logical Statement