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

factoids
Ð”Ð°ÐÐµÑ‚|Ð½ÐµÑ‚=no}}Ð”Ð°ÐÐµÑ‚|Ð½ÐµÑ‚=no}}Ð”Ð°ÐÐµÑ‚|Ð½ÐµÑ‚=no}}Ð”Ð°ÐÐµÑ‚|Ð½ÐµÑ‚=no}}}}File:Venn 0110 1001.svg|220px|thumb|rightExclusive or or exclusive disjunction is a logical operation that outputs true only when inputs differ (one is true, the other is false).WEB, Germundsson, Roger, Weisstein, Eric, XOR,weblink MathWorld, Wolfram Research, 17 June 2015, It is symbolized by the prefix operator J{{citation|title=Routledge Encyclopedia of Philosophy|volume=10|editor-first=Edward|editor-last=Craig|publisher=Taylor & Francis|year=1998|isbn=9780415073103|page=496|url=https://books.google.com/books?id=HP9O6OM4iOgC&pg=PA496}} and by the infix operators XOR ({{IPAc-en|ËŒ|É›|k|s|_|Ëˆ|É”r}} or {{IPAc-en|Ëˆ|z|É”É¹}}), EOR, EXOR, âŠ», â©’, â©›, âŠ•, â†®, and . The negation of XOR is logical biconditional, which outputs true only when both inputs are the same.It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator excludes that case. This is sometimes thought of as "one or the other but not both". This could be written as "A or B, but not, A and B".More generally, XOR is true only when an odd number of inputs are true. A chain of XORsâ€”a XOR b XOR c XOR d (and so on)â€”is true whenever an odd number of the inputs are true and is false whenever an even number of inputs are true.

## Truth table

File:Multigrade operator XOR.svg|thumb|220px|Arguments on the left combined by XOR. This is a binary Walsh matrix (cf. Hadamard codeHadamard codeThe truth table of A XOR B shows that it outputs true whenever the inputs differ:{| class="wikitable" style="text-align:center"|+ XOR truth table!colspan="2" | Input || rowspan="2" | Output!A || B| 0| 1| 1| 0
• 0, false
• 1, true

## Equivalences, elimination, and introduction

Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is true if and only if one is true and the other is false. For example, if two horses are racing, then one of the two will win the race, but not both of them. The exclusive disjunction p oplus q, also denoted by p â©› q or operatorname{J}pq, can be expressed in terms of the logical conjunction ("logical and", wedge), the disjunction ("logical or", lor), and the negation (lnot) as follows:
begin{matrix}
p oplus q & = & (p lor q) land lnot (p land q)
end{matrix}The exclusive disjunction p oplus q can also be expressed in the following way:
begin{matrix}
p oplus q & = & (p land lnot q) lor (lnot p land q)
end{matrix}This representation of XOR may be found useful when constructing a circuit or network, because it has only one lnot operation and small number of wedge and lor operations. A proof of this identity is given below:
begin{matrix}
p oplus q & = & (p land lnot q) & lor & (lnot p land q) [3pt]
& = & ((p land lnot q) lor lnot p) & land & ((p land lnot q) lor q) [3pt]
& = & ((p lor lnot p) land (lnot q lor lnot p)) & land & ((p lor q) land (lnot q lor q)) [3pt]
& = & (lnot p lor lnot q) & land & (p lor q) [3pt]
& = & lnot (p land q) & land & (p lor q)
end{matrix}It is sometimes useful to write p oplus q in the following way:
begin{matrix}
p oplus q & = & lnot ((p land q) lor (lnot p land lnot q))
end{matrix}or:
begin{matrix}
p oplus q & = & (p lor q) land (lnot p lor lnot q)
end{matrix}This equivalence can be established by applying De Morgan's laws twice to the fourth line of the above proof.The exclusive or is also equivalent to the negation of a logical biconditional, by the rules of material implication (a material conditional is equivalent to the disjunction of the negation of its antecedent and its consequence) and material equivalence.In summary, we have, in mathematical and in engineering notation:
begin{matrix}
p oplus q & = & (p land lnot q) & lor & (lnot p land q) & = & poverline{q} + overline{p}q [3pt]
& = & (p lor q) & land & (lnot p lor lnot q) & = & (p + q)(overline{p} + overline{q}) [3pt]
& = & (p lor q) & land & lnot (p land q) & = & (p + q)(overline{pq})
end{matrix}

## Relation to modern algebra

Although the operators wedge (conjunction) and lor (disjunction) are very useful in logic systems, they fail a more generalizable structure in the following way:The systems ({T, F}, wedge) and ({T, F}, lor) are monoids, but neither is a group. This unfortunately prevents the combination of these two systems into larger structures, such as a mathematical ring.However, the system using exclusive or ({T, F}, oplus) is an abelian group. The combination of operators wedge and oplus over elements {T, F} produce the well-known field F_2. This field can represent any logic obtainable with the system (land, lor) and has the added benefit of the arsenal of algebraic analysis tools for fields.More specifically, if one associates F with 0 and T with 1, one can interpret the logical "AND" operation as multiplication on F_2 and the "XOR" operation as addition on F_2:
begin{matrix}
r = p land q & Leftrightarrow & r = p cdot q pmod 2 [3pt]
r = p oplus q & Leftrightarrow & r = p + q pmod 2
end{matrix}Using this basis to describe a boolean system is referred to as algebraic normal form.

## Exclusive "or" in English

{{essay|section|date=May 2013}}The Oxford English Dictionary explains "either ... or" as follows:
either of the two, but not both.or, conj.2 (adv.3) 2a Oxford English Dictionary, second edition (1989). OED Online.}}

## Alternative symbols

The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphasized in a given context of discussion. In addition to the abbreviation "XOR", any of the following symbols may also be seen:
• +, a plus sign, which has the advantage that all of the ordinary algebraic properties of mathematical rings and fields can be used without further ado; but the plus sign is also used for inclusive disjunction in some notation systems; note that exclusive disjunction corresponds to addition modulo 2, which has the following addition table, clearly isomorphic to the one above:
{| class="wikitable" style="margin-left: 20px; text-align: center"!  p !  q ! p+q| 0| 1| 1| 0
• oplus, a modified plus sign; this symbol is also used in mathematics for the direct sum of algebraic structures
• J, as in Jpq
• An inclusive disjunction symbol (lor) that is modified in some way, such as
• underlinelor
• dotvee
• ^, the caret, used in several programming languages, such as C, C++, C, D, Java, Perl, Ruby, PHP and Python, denoting the bitwise XOR operator; not used outside of programming contexts because it is too easily confused with other uses of the caret
• (File:X-or.svg|24px), sometimes written as
• '''>-

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