Negation Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy

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

x \ge 5

', (c) 'Every student passed.'

Solution

1
(a) 'It is not raining.'
2
(b) ' $x < 5$ ' (negating $\ge$ gives $<$ ).
3
(c) 'There exists a student who did not pass.' (negating $\forall$ gives $\exists \neg$ ).

Answer

(a)\;\text{Not raining.}\quad (b)\;x < 5.\quad (c)\;\text{Some student did not pass.}

For simple statements, add 'not'. For inequalities, flip the sign. For universal statements, negate by switching to an existential claim.

About Negation

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.

Learn more about Negation →

More Negation Examples

Example 1 easy

Write the negation of each statement and determine its truth value: (a) '[formula]', (b) 'All cats a

Example 2 medium

Simplify [formula] using logical laws.

Example 4 medium

Build the truth table for [formula] and verify that [formula] is always false (contradiction) and [f

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Related Concepts

Logical Statement