Disjonctif

Disjonctif, in logic and mathematics, refers to the disjunction operation, commonly denoted by the symbol ∨ and read as "or" in ordinary language. It combines two or more propositions to form a new proposition that is true if at least one of the components is true. For two propositions A and B, A ∨ B is true when A is true, or B is true, or both.

The truth-functional nature of disjunction is captured by a standard truth table: T ∨ T equals T, T ∨ F equals T, F ∨ T equals T, and F ∨ F equals F. Disjunction is commutative and associative, and it distributes over conjunction (A ∨ (B ∧ C) ≡ (A ∨ B) ∧ (A ∨ C)). De Morgan’s laws relate negation to disjunction: ¬(A ∨ B) ≡ ¬A ∧ ¬B, and ¬(A ∧ B) ≡ ¬A ∨ ¬B. In set theory, a disjunction corresponds to the union of sets when the propositions express membership conditions.

In practical terms, disjunction forms the backbone of many logical and computational frameworks. In programming languages, the boolean OR operator implements short-circuit evaluation: if the first operand is true, the second may not be evaluated. Disjunctive normal form expresses a formula as a disjunction of conjunctions of literals, while clauses in conjunctive normal form are disjunctions of literals. These forms facilitate logical reasoning, simplification, and automated deduction in various disciplines.