Negation Formula

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

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

Notation

\neg P or \sim P or 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.'

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.

Common Mistakes

  • Negating 'All X are Y' as 'No X are Y' instead of 'Some X are not Y'
  • Thinking negation changes a statement's subject — \neg P just flips the truth value, it doesn't create a 'stronger opposite'
  • Forgetting double negation cancels out — \neg(\neg P) = P, not something new

Why This Formula Matters

Essential for expressing opposites and proof by contradiction.

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?

Essential for expressing opposites and proof by contradiction.

What do students get wrong about Negation?

Negation of 'All dogs bark' is 'Some dog doesn't bark,' not 'No dogs bark.'

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