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In logic and mathematics, or, also known as logical disjunction or inclusive disjunction is a logical operator that results in true whenever one or more of its operands are true. In grammar, or is a coordinating conjunction. In ordinary language "or" rather has the meaning of exclusive disjunction.

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. 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.

p | q | ∨ |
---|---|---|

T | T | T |

T | F | T |

F | T | T |

F | F | F |

The following properties apply to disjunction:

- associativity: $a\; lor\; (b\; lor\; c)\; equiv\; (a\; lor\; b)\; lor\; c$
- commutativity: $a\; lor\; b\; equiv\; b\; lor\; a$
- distributivity: $(a\; lor\; (b\; land\; c))\; equiv\; ((a\; lor\; b)\; land\; (a\; lor\; c))$

- $(a\; land\; (b\; lor\; c))\; equiv\; ((a\; land\; b)\; lor\; (a\; land\; c))$

- $(a\; lor\; (b\; equiv\; c))\; equiv\; ((a\; lor\; b)\; equiv\; (a\; lor\; c))$

- idempotency: $a\; lor\; a\; equiv\; a$
- monotonicity: $(a\; rightarrow\; b)\; rightarrow\; ((c\; lor\; a)\; rightarrow\; (c\; lor\; b))$

- $(a\; rightarrow\; b)\; rightarrow\; ((a\; lor\; c)\; rightarrow\; (b\; lor\; c))$

- 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", the symbol "$or$", deriving from the Latin word vel for "or", is commonly used for disjunction. For example: "A $or$ 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:

- $A\; or\; B$

- $neg\; A\; or\; B$

- $A\; or\; neg\; B\; or\; neg\; C\; or\; D\; or\; neg\; E$

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

- 0 or 0 = 0
- 0 or 1 = 1
- 1 or 0 = 1
- 1 or 1 = 1
- 1010 or 1100 = 1110

The `or`

operator can be used to set bits in a bitfield to 1, by `or`

-ing the field with a constant field with the relevant bits set to 1.

`|`

) and logical disjunction with the double pipe (`||`

) operators.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.

Although in most languages the type of a logical disjunction expression is boolean and thus can only have the value `true`

or `false`

, in some (such as Python and JavaScript) the logical disjunction operator returns one of its operands; the first operand if it evaluates to a true value, and the second operand otherwise.

- 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.

- Exclusive disjunction
- Affirming a disjunct
- Bitwise OR
- Boolean algebra (logic)
- Boolean algebra topics
- Boolean domain

- Boolean function
- Boolean-valued function
- Disjunctive syllogism
- Disjunction elimination
- Disjunction introduction
- First-order logic

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

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Last updated on Friday October 10, 2008 at 12:44:13 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday October 10, 2008 at 12:44:13 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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