De Morgan's Laws — Definition, Formula & Examples

De Morgan's Laws are two rules that show how to negate a compound logical statement. The negation of an AND becomes an OR (with each part negated), and the negation of an OR becomes an AND (with each part negated).

For propositions $p$ and $q$ : (1) $\neg(p \land q) \equiv \neg p \lor \neg q$ , and (2) $\neg(p \lor q) \equiv \neg p \land \neg q$ . These equivalences hold in propositional logic and extend to set theory, where complement distributes over union and intersection analogously.

Key Formula

\neg(p \land q) \equiv \neg p \lor \neg q \qquad \neg(p \lor q) \equiv \neg p \land \neg q

Where:

$p, q$ = Propositions (statements that are either true or false)
$\neg$ = Negation (NOT)
$\land$ = Conjunction (AND)
$\lor$ = Disjunction (OR)
$\equiv$ = Logically equivalent

How It Works

When you negate a statement that uses AND or OR, De Morgan's Laws tell you to do three things: negate each individual part, and swap the connector (AND becomes OR, OR becomes AND). For example, the negation of "it is raining and it is cold" is "it is not raining or it is not cold." These laws work the same way in set theory: the complement of

A \cap B

equals

\overline{A} \cup \overline{B}

, and the complement of

A \cup B

equals

\overline{A} \cap \overline{B}

Worked Example

Problem: Write the negation of the statement: "x > 3 and x ≤ 10."

Identify the structure: The statement has the form

p \land q

, where

p

is "

x > 3

" and

q

is "

x \leq 10

Apply De Morgan's first law: Negate each part and change AND to OR.

\neg(p \land q) \equiv \neg p \lor \neg q

Write the result: Negate each proposition:

\neg p

is "

x \leq 3

" and

\neg q

is "

x > 10

x \leq 3 \;\lor\; x > 10

Answer: The negation is "

x \leq 3

x > 10

Why It Matters

De Morgan's Laws appear constantly in proof writing, programming (simplifying Boolean conditions in if-statements), and circuit design. In discrete math and theoretical computer science courses, they are essential tools for rewriting logical expressions into equivalent forms.

Common Mistakes

Mistake: Negating the compound statement but forgetting to swap AND with OR (or vice versa).

Correction: Always change the connector when you distribute the negation. Negating both parts alone is not enough —

\neg(p \land q)

\neg p \lor \neg q

, not

\neg p \land \neg q