Logical disjunction

Logical disjunction

OR
Definition	${\displaystyle x+y}$
Truth table	${\displaystyle (1110)}$
Logic gate
Normal forms
Disjunctive	${\displaystyle x+y}$
Conjunctive	${\displaystyle x+y}$
Zhegalkin polynomial	${\displaystyle x\oplus y\oplus xy}$
Post's lattices
0-preserving	yes
1-preserving	yes
Monotone	yes
Affine	no
Self-dual	no
v t e

Logical connectives
AND ${\displaystyle A\land B}$, ${\displaystyle A\cdot B}$, ${\displaystyle AB}$, ${\displaystyle A\&B}$, ${\displaystyle A\&\&B}$ equivalent ${\displaystyle A\equiv B}$, ${\displaystyle A\Leftrightarrow B}$, ${\displaystyle A\leftrightharpoons B}$ implies ${\displaystyle A\Rightarrow B}$, ${\displaystyle A\supset B}$, ${\displaystyle A\rightarrow B}$ NAND ${\displaystyle A{\overline {\land }}B}$, ${\displaystyle A\uparrow B}$, ${\displaystyle A\mid B}$, ${\displaystyle {\overline {A\cdot B}}}$ nonequivalent ${\displaystyle A\not \equiv B}$, ${\displaystyle A\not \Leftrightarrow B}$, ${\displaystyle A\nleftrightarrow B}$ NOR ${\displaystyle A{\overline {\lor }}B}$, ${\displaystyle A\downarrow B}$, ${\displaystyle {\overline {A+B}}}$ NOT ${\displaystyle \neg A}$, ${\displaystyle -A}$, ${\displaystyle {\overline {A}}}$, ${\displaystyle \sim A}$ OR ${\displaystyle A\lor B}$, ${\displaystyle A+B}$, ${\displaystyle A\mid B}$, ${\displaystyle A\parallel B}$ XNOR ${\displaystyle A\ {\text{XNOR}}\ B}$ XOR ${\displaystyle A{\underline {\lor }}B}$, ${\displaystyle A\oplus B}$ converse ${\displaystyle A\Leftarrow B}$, ${\displaystyle A\subset B}$, ${\displaystyle A\leftarrow B}$
Related concepts
Propositional calculus Predicate logic Boolean algebra Truth table Truth function Boolean function Functional completeness Scope (logic)
Applications
Digital logic Programming languages Mathematical logic Philosophy of logic
Category
v t e

In logic, disjunction, also known as logical disjunction or logical or or logical addition or inclusive disjunction, is a logical connective typically notated as ${\displaystyle \lor }$ and read aloud as "or". For instance, the English language sentence "it is sunny or it is warm" can be represented in logic using the disjunctive formula ${\displaystyle S\lor W}$, assuming that ${\displaystyle S}$ abbreviates "it is sunny" and ${\displaystyle W}$ abbreviates "it is warm".

In classical logic, disjunction is given a truth functional semantics according to which a formula ${\displaystyle \phi \lor \psi }$ is true unless both ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are false. Because this semantics allows a disjunctive formula to be true when both of its disjuncts are true, it is an inclusive interpretation of disjunction, in contrast with

An operand of a disjunction is a disjunct.^[3]

Because the logical "or" means a disjunction formula is true when either one or both of its parts are true, it is referred to as an inclusive disjunction. This is in contrast with an exclusive disjunction, which is true when one or the other of the arguments are true, but not both (referred to as "exclusive or", or "XOR").

When it is necessary to clarify whether inclusive or exclusive "or" is intended, English speakers sometimes uses the phrase "and/or". In terms of logic, this phrase is identical to "or", but makes the inclusion of both being true explicit.

In logic and related fields, disjunction is customarily notated with an infix operator ${\displaystyle \lor }$ (Unicode U+2228 ∨ LOGICAL OR).^[1] Alternative notations include ${\displaystyle +}$, used mainly in electronics, as well as ${\displaystyle \vert }$ and ${\displaystyle \vert \!\vert }$ in many programming languages. The English word "or" is sometimes used as well, often in capital letters. In Jan Łukasiewicz's prefix notation for logic, the operator is ${\displaystyle A}$, short for Polish alternatywa (English: alternative).^[4]

In mathematics, the disjunction of an arbitrary number of elements ${\displaystyle a_{1},\ldots ,a_{n}}$ can be denoted as an iterated binary operation using a larger ⋁ (Unicode U+22C1 ⋁ N-ARY LOGICAL OR):^[5]

${\displaystyle \bigvee _{i=1}^{n}a_{i}=a_{1}\lor a_{2}\lor \ldots a_{n-1}\lor a_{n}}$

In the

This semantics corresponds to the following truth table:^[1]

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle A\lor B}$
F	F	F
F	T	T
T	F	T
T	T	T

In classical logic systems where logical disjunction is not a primitive, it can be defined in terms of the primitive "and" (${\displaystyle \land }$) and "

${\displaystyle \lnot }$

${\displaystyle A\lor B=\neg ((\neg A)\land (\neg B))}$.

Alternatively, it may be defined in terms of "implies" (${\displaystyle \to }$) and "not" as:^[7]

${\displaystyle A\lor B=(\lnot A)\to B}$.

The latter can be checked by the following truth table:

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle \neg A}$	${\displaystyle \neg A\rightarrow B}$	${\displaystyle A\lor B}$
F	F	T	F	F
F	T	T	T	T
T	F	F	T	T
T	T	F	T	T

It may also be defined solely in terms of ${\displaystyle \to }$:

${\displaystyle A\lor B=(A\to B)\to B}$.

It can be checked by the following truth table:

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle A\rightarrow B}$	${\displaystyle (A\rightarrow B)\rightarrow B}$	${\displaystyle A\lor B}$
F	F	T	F	F
F	T	T	T	T
T	F	F	T	T
T	T	T	T	T

The following properties apply to disjunction:

• Associativity
  : ${\displaystyle a\lor (b\lor c)\equiv (a\lor b)\lor c}$^[8]
• Commutativity
  : ${\displaystyle a\lor b\equiv b\lor a}$
• Distributivity
  : ${\displaystyle (a\land (b\lor c))\equiv ((a\land b)\lor (a\land c))}$

${\displaystyle (a\lor (b\land c))\equiv ((a\lor b)\land (a\lor c))}$

${\displaystyle (a\lor (b\lor c))\equiv ((a\lor b)\lor (a\lor c))}$

${\displaystyle (a\lor (b\equiv c))\equiv ((a\lor b)\equiv (a\lor c))}$

• Idempotency
  : ${\displaystyle a\lor a\equiv a}$
• Monotonicity
  : ${\displaystyle (a\rightarrow b)\rightarrow ((c\lor a)\rightarrow (c\lor b))}$

${\displaystyle (a\rightarrow b)\rightarrow ((a\lor c)\rightarrow (b\lor c))}$

• 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.

This section needs expansion. You can help by adding to it. (February 2021)

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 1100 = 1110

The or operator can be used to set bits in a bit field to 1, by or-ing the field with a constant field with the relevant bits set to 1. For example, x = x | 0b00000001 will force the final bit to 1, while leaving other bits unchanged.^{[citation needed]}

Many languages distinguish between bitwise and logical disjunction by providing two distinct operators; in languages following

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.

In a parallel (concurrent) language, it is possible to short-circuit both sides: they are evaluated in parallel, and if one terminates with value true, the other is interrupted. This operator is thus called the parallel or.

Although the type of a logical disjunction expression is Boolean in most languages (and thus can only have the value true or false), in some languages (such as

Set theory

This section needs expansion. You can help by adding to it. (February 2021)

The membership of an element of a union set in set theory is defined in terms of a logical disjunction: ${\displaystyle x\in A\cup B\Leftrightarrow (x\in A)\vee (x\in B)}$. Because of this, logical disjunction satisfies many of the same identities as set-theoretic union, such as

Disjunction in natural languages does not precisely match the interpretation of ${\displaystyle \lor }$ in classical logic. Notably, classical disjunction is inclusive while natural language disjunction is often understood exclusively, as the following English example typically would be.^[1]

• Mary is eating an apple or a pear.

This inference has sometimes been understood as an

• George 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.