Logical disjunction
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In logic and mathematics, logical disjunction (usual symbol or) is a logical operator that results in true if either of the operands is true.
Informally speaking, 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", for example, "would you like tea or coffee?". In logic, this is called an exclusive disjunction or an 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.1 (An everyday example of this meaning would be "damage caused by scratches or dents is chargeable"—since presumably damage caused by scratches AND dents is therefore also chargeable).
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:
| p | q | p ∨ q |
|---|---|---|
| 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.
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:
- A ∨ B
- ¬A ∨ B
- A ∨ ¬B ∨ ¬C ∨ D ∨ ¬E
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)
The operator xor is also commutative and therefore the order of the operands is not important:
- A or B ⇔ 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