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[[Image:ORGate2.png|thumb|right|150px|OR [[logic gate]].]]
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In [[logic]] and [[mathematics]], '''logical disjunction''' (written '''''or''''') is a [[logical operator]] that results in true just whenever '''some''' of its operands are true.

==Definition==

'''Logical disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' if and only if both of its operands are false.

The [[truth table]] of '''p OR q''' (also written as '''p &or; q''') is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Logical Disjunction'''
|- style="background:paleturquoise"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p &or; q
|-
| F || F || F
|-
| F || T || T
|-
| T || F || T
|-
| T || T || T
|}
<br>

More generally a disjunction is a logical formula that can have one or more [[literal]]s separated only by ORs. A single literal is often considered to be a degenerate disjunction.

==Symbol==

The mathematical symbol for logical disjunction varies in the literature.  In addition to the word "or", the symbol "&or;", deriving from the Latin word ''vel'' for "or", is commonly used for disjunction.  For example: "''A'' &or; ''B''&nbsp;" is read as "''A'' or ''B''&nbsp;".  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:
: ''A'' &or; ''B''
: &not;''A'' &or; ''B''
: ''A'' &or; &not;''B'' &or; &not;''C'' &or; ''D'' &or; &not;''E''

The corresponding operation in set theory is the [[union (set theory)|set-theoretic union]].

==Algebraic properties==

For more than two inputs, ''OR'' can be applied to the first two inputs, and then the result can be ''OR'ed'' with each subsequent input:

: (''A'' or (''B'' or ''C'')) &hArr; ((''A'' or ''B'') or ''C'')

Because OR is [[Associativity|associative]], the order of the inputs does not matter:  the same result will be obtained regardless of association.

The operator OR is also [[Commutativity|commutative]] and therefore the order of the operands is not important:

: ''A'' or ''B''  &hArr; ''B'' or ''A''

==Bitwise operation==

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 1110 = 1110

Note that in computer science the OR operator can be used to set a [[bit]] to 1 by OR-ing the bit with 1.

==Union==

The [[union (set theory)|union]] used in [[set theory]] is defined in terms of a logical disjunction: ''x'' &#8712; ''A'' &#8746; ''B'' if and only if (''x'' &#8712; ''A'') &#8744; (''x'' &#8712; ''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]].

==Notes==

* [[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.

==See also==

===Logical operators===
{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Negation]]
{{col-end}}

===Related topics===
{{col-begin}}
{{col-break}}
* [[Affirming a disjunct]]
* [[Boolean algebra]]
* [[Boolean algebra topics]]
* [[Boolean domain]]
* [[Boolean function]]
{{col-break}}
* [[Boolean logic]]
* [[Boolean-valued function]]
* [[Disjunctive syllogism]]
* [[First-order logic]]
* [[Logical graph]]
{{col-break}}
* [[Logical value]]
* [[Operation (mathematics)|Operation]]
* [[Operator]]
* [[Propositional logic]]
* [[Zeroth order logic]]
{{col-end}}

==External links==

* [http://plato.stanford.edu/entries/disjunction/ Stanford Encyclopedia of Philosophy entry]
* [http://mathworld.wolfram.com/Disjunction.html Eric W. Weisstein. "Disjunction." From MathWorld--A Wolfram Web Resource]

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* Adapted from the [[Wikipedia]] article, "Logical_disjunction" [http://en.wikipedia.org/wiki/Logical_disjunction http://en.wikipedia.org/wiki/Logical_disjunction], used under the [[GNU Free Documentation License]].

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