Negation Formula

Q: What is the Negation formula?

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.

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

The Formula

\neg(\neg P) \Leftrightarrow P

(double negation law)

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

Quick Example

P

is 'It is raining' (T), then

\sim P

is 'It is not raining' (F).

Notation

\neg P

\sim P

P'

What This Formula Means

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...'

Formal View

\neg P \Leftrightarrow (P \to \bot)

;

\neg(\neg P) \Leftrightarrow P

(double negation);

\neg(\forall x\,P(x)) \Leftrightarrow \exists x\,\neg P(x)

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

Common Mistakes

Negating 'all are' as 'none are' — the negation of $\forall x\,P(x)$ is $\exists x\,\neg P(x)$ , 'at least one is not.'
Treating an extreme opposite as a negation — $\neg(\text{tall})$ is 'not tall', not 'short.'
Mishandling double negation — $\neg(\neg P)$ returns to $P$ , not something stronger.

Why This Formula Matters

Negation is the NOT of logic and the engine of indirect proof and De Morgan's laws. A student who negates 'all are' to 'all are not' (instead of 'at least one is not'), or who double-negates wrongly, derives false 'opposites' that wreck proofs and quantifier work. Recognizing it by "Is this new statement true in exactly the cases where the original is false?" — rather than by familiar numbers — is what lets a student tell it apart from opposite/contrary statement and converse and complement (sets) in a mixed problem set.

Frequently Asked Questions

What is the Negation formula?

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.

How do you use the Negation formula?

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

What do the symbols mean in the Negation formula?

$\neg P$ or $\sim P$ or $P'$

Why is the Negation formula important in Math?

What do students get wrong about Negation?

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.

What should I learn before the Negation formula?

Before studying the Negation formula, you should understand: logical statement.

← Back to Negation See All Examples Practice Problems

Related Concepts

Logical Statement

Related Formulas

Logical Statement Formula