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Logical disjunction
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Logical disjunction

   

In logic and mathematics, a disjunction is an "or statement". For example "John skis or Sally swims" is a disjunction.

Note that in everyday language, use of the word "or" can sometimes mean "either, but not both" (eg, "would you like tea or coffee?"). In logic, this is called an "exclusive disjunction or "exclusive or". When used formally, "or" allows for both parts of the or statement (its disjuncts) to be true ("and/or"), therefore "or" is also called inclusive disjunction. Note: 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.

The statement "P or Q" is often written as

P Q
Such a disjunction is false if both P and Q are false. In all other cases it is true.

For two inputs A and B, the truth table of the function is as follows.

A B | A or B
----+--------
F F |    F
F T |    T
T F |    T
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.

For example, all the following are disjunctions:

A B
¬A B
A ¬B ¬C D ¬E

The equivalent notion in set theory is the set theoretic union.

See also