Exclusive or

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Exclusive or

XOR
Truth table	${\displaystyle (0110)}$
Logic gate
Normal forms
Disjunctive	${\displaystyle {\overline {x}}\cdot y+x\cdot {\overline {y}}}$
Conjunctive	${\displaystyle ({\overline {x}}+{\overline {y}})\cdot (x+y)}$
Zhegalkin polynomial	${\displaystyle x\oplus y}$
Post's lattices
0-preserving	yes
1-preserving	no
Monotone	no
Affine	yes
Self-dual	no
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Logical connectives
NOT ${\displaystyle \neg A,-A,{\overline {A}},\sim A}$ AND ${\displaystyle A\land B,A\cdot B,AB,A\ \&\ B,A\ \&\&\ B}$ NAND ${\displaystyle A{\overline {\land }}B,A\uparrow B,A\mid B,{\overline {A\cdot B}}}$ OR ${\displaystyle A\lor B,A+B,A\mid B,A\parallel B}$ NOR ${\displaystyle A{\overline {\lor }}B,A\downarrow B,{\overline {A+B}}}$ XNOR ${\displaystyle A\odot B,{\overline {A{\overline {\lor }}B}}}$ └ equivalent ${\displaystyle A\equiv B,A\Leftrightarrow B,A\leftrightharpoons B}$ XOR ${\displaystyle A{\underline {\lor }}B,A\oplus B}$ └nonequivalent ${\displaystyle A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B}$ implies ${\displaystyle A\Rightarrow B,A\supset B,A\rightarrow B}$ nonimplication (NIMPLY) ${\displaystyle A\not \Rightarrow B,A\not \supset B,A\nrightarrow B}$ converse ${\displaystyle A\Leftarrow B,A\subset B,A\leftarrow B}$ converse nonimplication ${\displaystyle A\not \Leftarrow B,A\not \subset B,A\nleftarrow B}$
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
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Exclusive or, exclusive disjunction, exclusive alternation, logical non-equivalence, or logical inequality is a logical operator whose negation is the logical biconditional. With two inputs, XOR is true if and only if the inputs differ (one is true, one is false). With multiple inputs, XOR is true if and only if the number of true inputs is odd.^[1]

It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true. XOR excludes that case. Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B".

It is symbolized by the prefix operator ${\displaystyle J}$^[2]: 16 and by the infix operators XOR (/ˌɛks ˈɔːr/, /ˌɛks ˈɔː/, /ˈksɔːr/ or /ˈksɔː/), EOR, EXOR, ${\displaystyle {\dot {\vee }}}$, ${\displaystyle {\overline {\vee }}}$, ${\displaystyle {\underline {\vee }}}$, ⩛, ${\displaystyle \oplus }$, ${\displaystyle \nleftrightarrow }$, and ${\displaystyle \not \equiv }$.