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Negation Sign — Definition, Formula & Examples

The negation sign is the symbol ¬\neg (or sometimes ∼\sim) placed before a logical statement to reverse its truth value. If a statement is true, its negation is false, and vice versa.

Given a proposition pp, the negation ¬p\neg p is a unary logical operator defined such that ¬p\neg p is true if and only if pp is false. Its truth table assigns the opposite truth value to the operand.

Key Formula

¬p\neg p
Where:
  • ¬\neg = The negation operator, read as "not"
  • pp = Any proposition (a statement that is either true or false)

How It Works

Place the negation sign directly before the statement you want to negate. If pp represents "It is raining," then ¬p\neg p represents "It is not raining." When negating a compound statement, use parentheses carefully: ¬(p∧q)\neg(p \land q) negates the entire conjunction, whereas ¬p∧q\neg p \land q negates only pp. The truth table has just two rows: when pp is T, ¬p\neg p is F; when pp is F, ¬p\neg p is T.

Worked Example

Problem: Let p represent the statement "5 is even." Determine the truth value of ¬p.
Step 1: Evaluate the original statement p. The number 5 is odd, so p is false.
p=Fp = \text{F}
Step 2: Apply the negation sign. Since p is false, ¬p takes the opposite truth value.
¬p=T\neg p = \text{T}
Answer: ¬p is true, meaning "5 is not even" is a true statement.

Why It Matters

You need the negation sign every time you write a contrapositive, construct a proof by contradiction, or apply De Morgan's Laws. In computer science, negation maps directly to the NOT gate in circuit design and the `!` operator in programming languages.

Common Mistakes

Mistake: Negating a compound statement by distributing the sign incorrectly, e.g., writing ¬(p∧q)\neg(p \land q) as ¬p∧¬q\neg p \land \neg q.
Correction: By De Morgan's Law, ¬(p∧q)≡¬p∨¬q\neg(p \land q) \equiv \neg p \lor \neg q. The connective flips from AND to OR (and vice versa) when you distribute a negation.