From Textop Wiki
|The article below may contain errors of fact, bias, grammar, etc. The Citizendium Foundation and the participants in the Citizendium project make no representations about the reliability of this article or, generally, its suitability for any purpose. We make this disclaimer of all Citizendium article versions that have not been specifically approved.|
The truth table of p OR q (also written as p ∨ q) is as follows:
|p||q||p ∨ q|
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.
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:
- A ∨ B
- ¬A ∨ B
- A ∨ ¬B ∨ ¬C ∨ D ∨ ¬E
The corresponding operation in set theory is the set-theoretic union.
Associativity and commutativity
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)) ⇔ ((A or B) or C)
Because or is associative, the order of the inputs does not matter: the same result will be obtained regardless of association.
The operator xor is also commutative and therefore the order of the operands is not important:
- A or B ⇔ B or A
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.
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.
- 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.
- Portions of the above article are adapted from an earlier version of the Wikipedia article, "Logical disjunction", used under the GNU Free Documentation License.
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section titled GNU FDL text.