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{{redirect|XOR}}
{{for|the corresponding concept in [[combinational logic]]|XOR gate}}

'''Exclusive disjunction''', also known as '''exclusive or''' and symbolized by '''XOR''' or '''EOR''', is a [[logical operation]] on two operands that results in a [[logical value]] of ''true'' if and only if one of the operands, but not both, has a value of ''true''.

[[Image:XOR.jpg|thumb|right|150px|XOR [[Logic Gate]]]]

==Definition==

In many natural languages, [[English language|English]] included, the interpretation of the word "or" requires a certain amount of care.  The '''exclusive disjunction''' of a pair of propositions, (''p'', ''q''), means that ''p'' is true or ''q'' is true, but not both.  For example, the normal intention of a statement like "You can follow the rules or be disqualified" is to stipulate that exactly one of the conditions can be true.  In logic, however, the default meaning of the word "or" is the ''[[logical disjunction|inclusive disjunction]]'', which signifies that at least one of the alternatives is true.  Other languages, such as [[Latin]], may have different words for these different types of "or".

'''Exclusive disjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' just in case exactly one of its operands is true.

The [[truth table]] of '''p XOR q''' (also written as '''p + q''', '''p &oplus; q''', or '''p &ne; q''') is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Exclusive Disjunction'''
|- style="background:paleturquoise"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p XOR q
|-
| F || F || F
|-
| F || T || T
|-
| T || F || T
|-
| T || T || F
|}
<br>

The following equivalents can then be deduced:

: <math>\begin{matrix}
p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
\\
      & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\
\\
      & = & (p \lor q) & \land & \lnot (p \land q)
\end{matrix}</math>

'''Generalized''' or '''n-ary''' XOR is true when the number of 1-bits  is odd.

==Alternative symbols==
The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphaisized in a given context of discussion.  In addition to the abbreviation "XOR", any of the following symbols may also be seen:
* A plus sign (+).  This makes sense mathematically because exclusive disjunction corresponds to [[addition]] [[modular arithmetic|modulo]] 2, which has the following addition table, clearly [[isomorphism|isomorphic]] to the one above:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ '''Addition Modulo 2'''
|- style="background:paleturquoise"
! style="width:15%" | p
! style="width:15%" | q
! style="width:15%" | p + q
|-
| 0 || 0 || 0
|-
| 0 || 1 || 1
|-
| 1 || 0 || 1
|-
| 1 || 1 || 0
|}
<br>

* The use of the plus sign has the added advantage that all of the ordinary algebraic properties of mathematical [[ring (mathematics)|rings]] and [[field (mathematics)|fields]] can be used without further ado.
* A plus sign that is modified in some way, such as being encircled (&#8853;).  This usage faces the objection that this same symbol is already used in mathematics for the ''[[direct sum]]'' of algebraic structures.
* An inclusive disjunction symbol (&or;) that is modified in some way, such as being underlined (<u>&or;</u>).
* In the [[C programming language]] and the [[Java programming language]], a caret (^) is used to denote both the ordinary XOR and the bitwise XOR operator.  This is not used outside of programming contexts because it is too easily confused with other uses of the caret.

==Equivalent expressions==

The exclusive disjunction <math>p + q\!</math> can be expressed in terms of the conjunction (&and;), the disjunction (&or;), and the negation (&not;) as follows:

: <math>\begin{matrix}
p + q & = & (p \land \lnot q) \lor (\lnot p \land q)
\end{matrix}</math>

The exclusive disjunction <math>p + q\!</math> can also be expressed in the following way:

: <math>\begin{matrix}
p + q & = & \lnot (p \land q) \land (p \lor q)
\end{matrix}</math>

This representation of XOR may be found useful when constructing a circuit or network, because it has only one &not; operation and small number of &and; and &or; operations. The proof of this identity is given below:

: <math>\begin{matrix}
p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\
& = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\
& = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\
& = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\
& = & \lnot (p \land q) & \land & (p \lor q)
\end{matrix}</math>

It is sometimes useful to write ''p'' XOR ''q'' in the following way:

: <math>\begin{matrix}
p + q & = & \lnot ((p \land q) \lor (\lnot p \land \lnot q))
\end{matrix}</math>

This equivalence can be established by applying [[De Morgan's laws]] twice to the fourth line of the above proof.

==Associativity and commutativity==

In view of the [[isomorphism]] between addition modulo 2 and exclusive disjunction, it is clear that XOR is both an [[associative]] and a [[commutative]] operation.  Thus parentheses may be omitted in successive operations and the order of terms makes no difference to the result.  For example, we have the following equations:


: <math>\begin{matrix}
p + q & = & q + p \\
\\
(p + q) + r & = & p + (q + r) & = & p + q + r
\end{matrix}</math>

==Properties==
This section uses the following symbols:

: <math>\begin{matrix}
0         & = & \mbox{false}     \\
1         & = & \mbox{true}      \\
\lnot p   & = & \mbox{not}\ p    \\
p + q     & = & p\ \mbox{xor}\ q \\
p \land q & = & p\ \mbox{and}\ q \\
p \lor  q & = & p\ \mbox{or} \ q
\end{matrix}</math>

The following equations follow from logical axioms:

: <math>\begin{matrix}
p + 0       & = & p       \\
p + 1       & = & \lnot p \\
p + p       & = & 0       \\
p + \lnot p & = & 1       \\
\\
p + q         & = & q + p              \\
p + q + p     & = & q                  \\
p + (q + r)   & = & (p + q) + r        \\
p + q         & = & \lnot p + \lnot q  \\
\lnot (p + q) & = & \lnot p + q        & = & p + \lnot q \\
\\
p + (\lnot p \land q)      & = & p \lor  q       \\
p + (p \land \lnot q)      & = & p \land q       \\
p + (p \lor q)             & = & \lnot p \land q \\
\lnot p + (p \lor \lnot q) & = & p \lor q        \\
p \land (p + \lnot q)      & = & p \land q       \\
p \lor (p + q)             & = & p \lor q
\end{matrix}</math>

==Bitwise operation==
Exclusive disjunction is often used for bitwise operations. Examples:

* 1 xor 1 = 0
* 1 xor 0 = 1
* 1110 xor 1001 = 0111 (this is equivalent to addition without [[carry]])

As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of two ''n''-bit strings is identical to the standard vector of addition in the [[vector space]] <math>(\Z/2\Z)^n</math>.

==Computer science==

In computer science, exclusive disjunction has several uses:

* It tells whether two bits are unequal.
* It is an optional [[negation|bit-flipper]] (the deciding input chooses whether to invert the data input).
* It tells whether there are an [[even and odd numbers|odd]] number of 1&nbsp;bits (A &oplus; B &oplus; C &oplus; D &oplus; E is true [[iff]] an odd number of the variables are true).

On some computer architectures, it is more [[algorithmic efficiency|efficient]] to store a zero in a register by xor-ing the register with itself (bits xor-ed with themselves are always zero) instead of loading and storing the value zero. In simple threshold activated [[neural network]]s, modelling the 'xor' function requires a second layer because 'xor' is not a linearly-separable function.  

Exclusive-or is sometimes used as a simple mixing function in [[cryptography]], for example, with [[one-time pad]] or [[Feistel network]] systems.

XOR is used in [[Redundant array of independent disks|RAID]] 3&ndash;6 for creating parity information.  For example, RAID can "back up" bytes <code>10011100</code> and <code>01101100</code> from two (or more) hard drives by XORing (<code>11110000</code>) and writing to an another drive.  Under this method, if any one of the four hard drives are lost, the lost byte can be re-created by XORing bytes from the remaining drives.  If the drive containing <code>01101100</code> is lost, <code>10011100</code> and <code>11110000</code> can be XORed to recover the lost byte.

XOR is also used to detect an overflow in the result of a signed binary arithmetic operation. If the leftmost retained bit of the result is not the same as the infinite number of digits to the left, then that means overflow occurred. XORing those two bits will give a "one" if there is an overflow.

XOR can be used to [[XOR swap|swap]] two numeric variables in computers; however this is regarded as more of a curiosity and not encouraged in practice.

==See also==

===Logical operators===
{{col-begin}}
{{col-break}}
* [[Exclusive disjunction]]
* [[Logical conjunction]]
* [[Logical disjunction]]
* [[Logical equality]]
{{col-break}}
* [[Logical implication]]
* [[Logical NAND]]
* [[Logical NNOR]]
* [[Negation]]
{{col-end}}

===Related topics===
{{col-begin}}
{{col-break}}
* [[Affirming a disjunct]]
* [[Ampheck]]
* [[Boolean algebra]]
* [[Boolean algebra topics]]
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean logic]]
* [[Boolean-valued function]]
{{col-break}}
* [[Controlled NOT gate|Controlled NOT]]
* [[Disjunctive syllogism]]
* [[First-order logic]]
* [[Logical graph]]
* [[Logical value]]
* [[Minimal negation operator]]
* [[Multigrade operator]]
* [[Operation (mathematics)|Operation]]
{{col-break}}
* [[Operator]]
* [[Parametric operator]]
* [[Parity bit]]
* [[Propositional logic]]
* [[Symmetric difference]]
* [[XOR linked list]]
* [[XOR swap algorithm]]
* [[Zeroth order logic]]
{{col-end}}

[[Category:Logic]]
[[Category:Binary operations]]

[[de:Kontravalenz]]
[[fr:OU exclusif]]
[[it:Disgiunzione esclusiva]]
[[mk:&#1048;&#1089;&#1082;&#1083;&#1091;&#1095;&#1080;&#1090;&#1077;&#1083;&#1085;&#1072; &#1076;&#1080;&#1089;&#1112;&#1091;&#1085;&#1082;&#1094;&#1080;&#1112;&#1072;]]
[[nl:Exclusieve disjunctie]]
[[ja:&#25490;&#20182;&#30340;&#35542;&#29702;&#21644;]]
[[no:Eksklusiv disjunksjon]]
[[pl:Alternatywa wykluczaj&#261;ca]]
[[fi:XOR]]
[[tr:XOR kap&#305;s&#305;]]

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====References====

* Adapted from the [[Wikipedia]] article, "Exclusive_disjunction" [http://en.wikipedia.org/wiki/Exclusive_disjunction http://en.wikipedia.org/wiki/Exclusive_disjunction], used under the [[GNU Free Documentation License]]