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: Negation flips a statement's truth value: if P is true, \neg P is false, and vice versa. Double negation cancels: \neg(\neg P) = P.

Common stuck point: Negation of 'All dogs bark' is 'Some dog doesn't bark,' not 'No dogs bark.'

Sense of Study hint: Write 'It is NOT the case that...' before the statement, then simplify. For 'all' statements, switch to 'there exists one that does not.'

Worked Examples

Example 1

easy
Write the negation of each statement and determine its truth value: (a) '5 > 3', (b) 'All cats are black.'

Solution

  1. 1
    Recall that the negation \neg P of a statement P is the statement that is true exactly when P is false.
  2. 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. 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.

Answer

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

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).
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Related Concepts

Logical Statement

Background Knowledge

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

logical statement