< Disjunction
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{{Spoken Wikipedia|Logical disjunction.ogg|2006-07-21}}

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 operation on two logical values, typically the values of two propositions, 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 ∨ q) is as follows:

|+ Logical Disjunction style="background:paleturquoise" ! style="width:15%" | p ! style="width:15%" | q ! style="width:15%" | p ∨ q

	| F
	| T
	| T
	| T

More generally a disjunction is a logical formula that can have one or more literals 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 "∨", deriving from the Latin word vel for "or", is commonly used for disjunction. For example: "A ∨ 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:

The corresponding operation in set theory is the 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:

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

The operator OR is also commutative and therefore the order of the operands is not important:

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 used 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.

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

|+  

Exclusive disjunction Logical conjunction Logical disjunction Logical equality

Logical implication Logical NAND Logical NNOR Negation

Related topics

|+  

Affirming a disjunct Boolean algebra Boolean algebra topics Boolean domain Boolean function

Boolean logic Boolean-valued function Disjunctive syllogism First-order logic Logical graph

Logical value Operation Operator Propositional calculus Zeroth order logic

External links

• Stanford Encyclopedia of Philosophy entry
• Eric W. Weisstein, "Disjunction", From MathWorld — A Wolfram Web Resource.

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Some content adapted from the Wikinfo article "Logical disjunction" under the GNU Free Documentation License.

Disjunkce Disjunktion Disjunktsioon Disyunción lógica Disjonction logique Logika disjungsi Disgiunzione inclusiva או (לוגיקה) Disjunkcija Логичка дисјункција Logische disjunctie 論理和 Inklusiv disjunksjon Alternatywa Disjunção lógica Disjunkcia (logika) Дисјункција Logisk disjunktion การเลือกเชิงตรรกศาสตร์ Диз'юнкція (логічна)

(last updated by Jon Awbrey, 6:54pm EDT - Sat, Apr 07 2007)