Editing Exclusive or (section)

{{short description|True when either but not both inputs are true}}
{{Redirect|XOR|the logic gate|XOR gate|other uses|XOR (disambiguation)}}
{{refimprove|date=May 2013}}

{{Infobox logical connective
| title        = Exclusive disjunction
| other titles = XOR
| Venn diagram = Venn0110.svg
| wikifunction = Z10237
| definition   =
| truth table  = <math>(0110)</math>
| logic gate   = XOR_ANSI.svg
| DNF          = <math>\overline{x} \cdot y + x \cdot \overline{y}</math>
| CNF          = <math>( \overline{x} + \overline{y} ) \cdot ( x + y )</math>
| Zhegalkin    = <math>x \oplus y </math>
| 0-preserving = yes
| 1-preserving = no
| monotone     = no
| affine       = yes
| self-dual    = no
}}
{{Logical connectives sidebar}}

[[File:Venn 0110 1001.svg|220px|[[Venn diagram]] of <math>A \oplus B \oplus C</math>|thumb|right]]
'''Exclusive or''', '''exclusive disjunction''', '''exclusive alternation''', '''logical non-equivalence''', or [[Logical_equality#Inequality|logical inequality]] is a [[Logical connective|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 [[Parity (mathematics)|odd]].<ref name=wolfram>{{cite web|last1=Germundsson|first1=Roger|last2=Weisstein|first2=Eric|title=XOR|url=http://mathworld.wolfram.com/XOR.html|website=[[MathWorld]]|publisher=[[Wolfram Research]]|access-date=17 June 2015}}</ref>

It gains the name "exclusive or" because the meaning of "or" is ambiguous when both [[operand]]s 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 [[Table of logic symbols|symbolized]] by the prefix operator <math>J</math><ref name="Bochenski1949">{{cite book |last1=Bocheński |first1=J. M. |title=Précis de logique mathématique |url=https://burjcdigital.urjc.es/bitstream/handle/10115/1425/PRECIS_DE_LOGIQUE_MATHEMATIQUE.pdf?sequence=1&isAllowed=y |date=1949 |location=The Netherlands |publisher=F. G. Kroonder, Bussum, Pays-Bas |language=French}} Translated as {{cite book |last1=Bocheński |first1=J. M. |translator-last=Bird |translator-first=O. |title=A Precis of Mathematical Logic |date=1959 |publisher=D. Reidel Publishing Company |url=https://archive.org/details/precisofmathemat0000boch/ |url-access=limited |doi=10.1007/978-94-017-0592-9 |location=Dordrecht, Holland|isbn=978-90-481-8329-6 }}</ref>{{rp|page=16}} and by the [[infix operator]]s '''XOR''' ({{IPAc-en|ˌ|ɛ|k|s|_|ˈ|ɔ:|r}}, {{IPAc-en|ˌ|ɛ|k|s|_|ˈ|ɔ:}}, {{IPAc-en|'|k|s|ɔ:|r}} or {{IPAc-en|'|k|s|ɔ:}}), '''EOR''', '''EXOR''', <math>\dot{\vee}</math>, <math>\overline{\vee}</math>, <math>\underline{\vee}</math>, '''<span style="font-size:120%;">⩛</span>''', <math>\oplus</math>, <math>\nleftrightarrow</math>, and <math>\not\equiv</math>.