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.
The truth table of [math]A \lor B[/math]:
|[math]A[/math]||[math]B[/math]||[math]A \lor B[/math]|
The following properties apply to disjunction:
- associativity: [math]a \lor (b \lor c) \equiv (a \lor b) \lor c [/math]
- commutativity: [math]a \lor b \equiv b \lor a [/math]
- distributivity: [math](a \lor (b \land c)) \equiv ((a \lor b) \land (a \lor c))[/math]
- [math](a \lor (b \lor c)) \equiv ((a \lor b) \lor (a \lor c))[/math]
- [math](a \lor (b \equiv c)) \equiv ((a \lor b) \equiv (a \lor c))[/math]
- idempotency: [math]a \lor a \equiv a [/math]
- monotonicity: [math](a \rightarrow b) \rightarrow ((c \lor a) \rightarrow (c \lor b))[/math]
- [math](a \rightarrow b) \rightarrow ((a \lor c) \rightarrow (b \lor c))[/math]
- 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 "[math]\lor[/math]", deriving from the Latin word vel (“either”, “or”) is commonly used for disjunction. For example: "A [math]\lor[/math] B " is read as "A or B ". 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:
- [math]A \lor B[/math]
- [math]\neg A \lor B[/math]
- [math]A \lor \neg B \lor \neg C \lor D \lor \neg E.[/math]
The corresponding operation in set theory is the set-theoretic union.
Applications in computer science
Operators corresponding to logical disjunction exist in most programming languages.
Disjunction is often used for bitwise operations. Examples:
- 0 or 0 = 0
- 0 or 1 = 1
- 1 or 0 = 1
- 1 or 1 = 1
- 1010 or 1100 = 1110
or operator can be used to set bits in a bit field to 1, by
or-ing the field with a constant field with the relevant bits set to 1. For example,
x = x | 0b00000001 will force the final bit to 1 while leaving other bits unchanged.
Many languages distinguish between bitwise and logical disjunction by providing two distinct operators; in languages following C, bitwise disjunction is performed with the single pipe (
|) and logical disjunction with the double pipe (
Logical disjunction is usually short-circuited; that is, if the first (left) operand evaluates to
true then the second (right) operand is not evaluated. The logical disjunction operator thus usually constitutes a sequence point.
In a parallel (concurrent) language, it is possible to short-circuit both sides: they are evaluated in parallel, and if one terminates with value true, the other is interrupted. This operator is thus called the parallel or.
Although in most languages the type of a logical disjunction expression is boolean and thus can only have the value
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" [exclusive-]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.
- Hazewinkel, Michiel, ed. (2001), "Disjunction", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/d033260
- Aloni, Maria. "Disjunction". in Zalta, Edward N.. Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/disjunction/.
- Eric W. Weisstein. "Disjunction." From MathWorld—A Wolfram Web Resource
Original source: https://en.wikipedia.org/wiki/ Logical disjunction. Read more