Logical disjunction

Or is usually expressed with an infix operator: in mathematics and logic, ∨;[1][2] in electronics, +; and in most programming languages, |, ||, or or. In Jan Łukasiewicz's prefix notation for logic, the operator is A, for Polish alternatywa (English: alternative).[4]

Logical disjunction is an operation on two logical values, typically the values of two propositions, that has a value of false if and only if both of its operands are false. More generally, a disjunction is a logical formula that can have one or more literals separated only by 'or's. A single literal is often considered to be a degenerate disjunction.

The disjunctive identity is false, which is to say that the or of an expression with false has the same value as the original expression. In keeping with the concept of vacuous truth, when disjunction is defined as an operator or function of arbitrary arity, the empty disjunction (OR-ing over an empty set of operands) is generally defined as false.

Truth table

The truth table of ${\displaystyle A\lor B}$:[2]

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle A\lor B}$
T	T	T
T	F	T
F	T	T
F	F	F

The following properties apply to disjunction:

• Associativity: ${\displaystyle a\lor (b\lor c)\equiv (a\lor b)\lor c}$
• Commutativity: ${\displaystyle a\lor b\equiv b\lor a}$
• Distributivity: ${\displaystyle (a\lor (b\land c))\equiv ((a\lor b)\land (a\lor c))}$

${\displaystyle (a\lor (b\lor c))\equiv ((a\lor b)\lor (a\lor c))}$

${\displaystyle (a\lor (b\equiv c))\equiv ((a\lor b)\equiv (a\lor c))}$

• Idempotency: ${\displaystyle a\lor a\equiv a}$
• Monotonicity: ${\displaystyle (a\rightarrow b)\rightarrow ((c\lor a)\rightarrow (c\lor b))}$

${\displaystyle (a\rightarrow b)\rightarrow ((a\lor c)\rightarrow (b\lor c))}$

• Truth-preserving: The interpretation under which all variables are assigned a truth value of 'true', produces a truth value of 'true' as a result of disjunction.
• Falsehood-preserving: The interpretation under which all variables are assigned a truth value of 'false', produces a truth value of 'false' as a result of disjunction.

The mathematical symbol for logical disjunction varies in the literature. In addition to the word "or", and the formula "Apq", the symbol "${\displaystyle \lor }$", deriving from the Latin word vel (“either”, “or”) is commonly used for disjunction. For example: "A ${\displaystyle \lor }$ B " is read as "A or B ".[1] Such a disjunction is false if both A and B are false. In all other cases, it is true.

All of the following are disjunctions:

${\displaystyle A\lor B}$

${\displaystyle \neg A\lor B}$

${\displaystyle A\lor \neg B\lor \neg C\lor D\lor \neg E.}$

The corresponding operation in set theory is the set-theoretic union.

OR logic gate

Operators corresponding to logical disjunction exist in most programming languages.

The membership of an element of a union set in set theory is defined in terms of a logical disjunction: x ∈ A ∪ B if and only if (x ∈ A) ∨ (x ∈ B). Because of this, logical disjunction satisfies many of the same identities as set-theoretic union, such as associativity, commutativity, distributivity, and de Morgan's laws, identifying logical conjunction with set intersection, logical negation with set complement.

As with other notions formalized in mathematical logic, the meaning of the natural-language coordinating conjunction or is closely related to—but different from—the logical or. For example, "Please ring me or send an email" likely means "do one or the other, but not both". On the other hand, "Her grades are so good that either she's very bright or she studies hard" does not exclude the possibility of both. In other words, in ordinary language "or" (even if used with "either") can mean either the inclusive "or", or the exclusive "or".

• George Boole, closely following analogy with ordinary mathematics, premised, as a necessary condition to the definition of "x + y", that x and y were mutually exclusive. Jevons, and practically all mathematical logicians after him, advocated, on various grounds, the definition of "logical addition" in a form which does not necessitate mutual exclusiveness.

Wikimedia Commons has media related to Logical disjunction.

• "Disjunction", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Aloni, Maria. "Disjunction". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
• Eric W. Weisstein. "Disjunction." From MathWorld—A Wolfram Web Resource