Disjunction

Disjunction is a logical connective representing the operation "or" between two propositions. In formal notation it is written as A ∨ B. By convention, disjunction is inclusive: A ∨ B is true if at least one of A or B is true, including the possibility that both are true.

A simple way to understand disjunction is through its truth table: A ∨ B is true in all cases except when both A and B are false. For example, the statement “It is raining or it is snowing” is true if it is raining, or it is snowing, or both.

There are two related notions. Inclusive disjunction allows both operands to be true simultaneously; exclusive disjunction, often denoted XOR, is true only when exactly one of A or B is true.

Disjunction has several standard properties. It is commutative (A ∨ B ≡ B ∨ A) and associative (A ∨ (B ∨ C) ≡ (A ∨ B) ∨ C). The identity element is false: A ∨ false ≡ A. It distributes over conjunction: A ∨ (B ∧ C) ≡ (A ∨ B) ∧ (A ∨ C). De Morgan’s laws relate negation and disjunction: ¬(A ∨ B) ≡ (¬A) ∧ (¬B).

In other areas, disjunction has analogues. In set theory, disjunction corresponds to the union of sets (A ∪ B). In probability, P(A ∪ B) = P(A) + P(B) − P(A ∩ B). In computing, OR is a boolean operator often implemented with short-circuit evaluation, where the second operand is evaluated only if the first is false.

Disjunction is a fundamental connective across logic, mathematics, and related disciplines, providing the formal basis for expressing alternatives and their combinations.