Negation Formula

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

¬(¬P)⇔P (double negation law)

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

Quick Example

If P is 'It is raining' (T), then ∼P is 'It is not raining' (F).

Notation

¬P or ∼P or P′

What This Formula Means

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.

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

Formal View

¬P⇔(P→⊥); ¬(¬P)⇔P (double negation); ¬(∀x P(x))⇔∃x ¬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≤3  (False),(b)  Some cat is not black  (True)

First step

1
Recall that the negation ¬P of a statement P is the statement that is true exactly when P is false.

Full solution

  1. 2
    (a) P: '5>3' (True). The negation reverses the inequality: ¬P: '5≤3' (False). (b) P: 'All cats are black' has form ∀x,P(x). Its negation is ∃x,¬P(x): 'There exists a cat that is not black.'
  2. 3
    Truth values: (a) ¬P is False because 5>3 is true. (b) ¬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 (∀), the negation is an existential statement (∃). The original and its negation always have opposite truth values.

Example 2

medium
Simplify ¬(¬p∨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 ∀x P(x) is ∃x ¬P(x), 'at least one is not.'
  • Treating an extreme opposite as a negation — ¬(tall) is 'not tall', not 'short.'
  • Mishandling double negation — ¬(¬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 ¬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?

¬P or ∼P or P′

Why is the Negation formula important in Math?

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.

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.

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Related Concepts

Logical Statement

Related Formulas

Logical Statement Formula