Exclusive or

Arguments on the left combined by XOR. This is a binary Walsh matrix (cf. Hadamard code).

The truth table of A XOR B shows that it outputs true whenever the inputs differ:

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle A\oplus B}$
False	False	False
False	True	True
True	False	True
True	True	False

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 ${\displaystyle p\nleftrightarrow q}$, also denoted by ${\displaystyle p}$ ? ${\displaystyle q}$ or ${\displaystyle \operatorname {J} pq}$, can be expressed in terms of the logical conjunction ("logical and", ${\displaystyle \wedge }$), the disjunction ("logical or", ${\displaystyle \lor }$), and the negation (${\displaystyle \lnot }$) as follows:

${\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land \lnot (p\land q)\end{matrix}}}$

The exclusive disjunction ${\displaystyle p\nleftrightarrow q}$ can also be expressed in the following way:

${\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)\lor (\lnot p\land q)\end{matrix}}}$

This representation of XOR may be found useful when constructing a circuit or network, because it has only one ${\displaystyle \lnot }$ operation and small number of ${\displaystyle \wedge }$ and ${\displaystyle \lor }$ operations. A proof of this identity is given below:

${\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\[3pt]&=&((p\land \lnot q)\lor \lnot p)&\land &((p\land \lnot q)\lor q)\\[3pt]&=&((p\lor \lnot p)\land (\lnot q\lor \lnot p))&\land &((p\lor q)\land (\lnot q\lor q))\\[3pt]&=&(\lnot p\lor \lnot q)&\land &(p\lor q)\\[3pt]&=&\lnot (p\land q)&\land &(p\lor q)\end{matrix}}}$

It is sometimes useful to write ${\displaystyle p\nleftrightarrow q}$ in the following way:

${\displaystyle {\begin{matrix}p\nleftrightarrow q&=&\lnot ((p\land q)\lor (\lnot p\land \lnot q))\end{matrix}}}$

or:

${\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land (\lnot p\lor \lnot q)\end{matrix}}}$

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:

${\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)&=&p{\overline {q}}+{\overline {p}}q\\[3pt]&=&(p\lor q)&\land &(\lnot p\lor \lnot q)&=&(p+q)({\overline {p}}+{\overline {q}})\\[3pt]&=&(p\lor q)&\land &\lnot (p\land q)&=&(p+q)({\overline {pq}})\end{matrix}}}$

The spirit of De Morgan's laws can be applied, we have:

$${\displaystyle \lnot (p\nleftrightarrow q)=\lnot p\nleftrightarrow q=p\nleftrightarrow \lnot q}$$

Although the operators ${\displaystyle \wedge }$ (conjunction) and ${\displaystyle \lor }$ (disjunction) are very useful in logic systems, they fail a more generalizable structure in the following way:

The systems ${\displaystyle (\{T,F\},\wedge )}$ and ${\displaystyle (\{T,F\},\lor )}$ 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 ${\displaystyle (\{T,F\},\oplus )}$ is an abelian group. The combination of operators ${\displaystyle \wedge }$ and ${\displaystyle \oplus }$ over elements ${\displaystyle \{T,F\}}$ produce the well-known field ${\displaystyle F_{2}}$. This field can represent any logic obtainable with the system ${\displaystyle (\land ,\lor )}$ and has the added benefit of the arsenal of algebraic analysis tools for fields.

More specifically, if one associates ${\displaystyle F}$ with 0 and ${\displaystyle T}$ with 1, one can interpret the logical "AND" operation as multiplication on ${\displaystyle F_{2}}$ and the "XOR" operation as addition on ${\displaystyle F_{2}}$:

${\displaystyle {\begin{matrix}r=p\land q&\Leftrightarrow &r=p\cdot q{\pmod {2}}\\[3pt]r=p\oplus q&\Leftrightarrow &r=p+q{\pmod {2}}\\\end{matrix}}}$

Using this basis to describe a boolean system is referred to as algebraic normal form.

Disjunction is often understood exclusively in natural languages. In English, the disjunctive word "or" is often understood exclusively, particularly when used with the particle "either". The English example below would normally be understood in conversation as implying that Mary is not both a singer and a poet.[3][4]

1. Mary is a singer or a poet.

However, disjunction can also be understood inclusively, even in combination with "either". For instance, the first example below shows that "either" can be felicitously used in combination with an outright statement that both disjuncts are true. The second example shows that the exclusive inference vanishes away under downward entailing contexts. If disjunction were understood as exclusive in this example, it would leave open the possibility that some people ate both rice and beans.[3]

2. Mary is either a singer or a poet or both.

3. Nobody ate either rice or beans.

Examples such as the above have motivated analyses of the exclusivity inference as pragmatic conversational implicatures calculated on the basis of an inclusive semantics. Implicatures are typically cancellable and do not arise in downward entailing contexts if their calculation depends on the Maxim of Quantity. However, some researchers have treated exclusivity as a bona fide semantic entailment and proposed nonclassical logics which would validate it.[3]

This behavior of English "or" is also found in other languages. However, many languages have disjunctive constructions which are robustly exclusive such as French soit... soit.[3]

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:

${\displaystyle p}$	${\displaystyle q}$	${\displaystyle p+q}$
0	0	0
0	1	1
1	0	1
1	1	0

• ${\displaystyle \oplus }$, 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 (${\displaystyle \lor }$) that is modified in some way, such as
  • ${\displaystyle {\underline {\lor }}}$
  • ${\displaystyle {\dot {\vee }}}$
• ^, 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
• , sometimes written as
  • ><
  • >-<
• =1, in IEC symbology

Commutativity: yes

${\displaystyle A\oplus B}$	${\displaystyle \Leftrightarrow }$	${\displaystyle B\oplus A}$
	${\displaystyle \Leftrightarrow }$

Associativity: yes

${\displaystyle ~A}$	${\displaystyle ~~~\oplus ~~~}$	${\displaystyle (B\oplus C)}$	${\displaystyle \Leftrightarrow }$			${\displaystyle (A\oplus B)}$	${\displaystyle ~~~\oplus ~~~}$	${\displaystyle ~C}$
	${\displaystyle ~~~\oplus ~~~}$		${\displaystyle \Leftrightarrow }$		${\displaystyle \Leftrightarrow }$		${\displaystyle ~~~\oplus ~~~}$

Distributivity:

The exclusive or doesn't distribute over any binary function (not even itself), but logical conjunction distributes over exclusive or. ${\displaystyle C\land (A\oplus B)=(C\land A)\oplus (C\land B)}$ (Conjunction and exclusive or form the multiplication and addition operations of a field GF(2), and as in any field they obey the distributive law.)

Idempotency: no

${\displaystyle ~A~}$	${\displaystyle ~\oplus ~}$	${\displaystyle ~A~}$	${\displaystyle \Leftrightarrow }$	${\displaystyle ~0~}$	${\displaystyle \nLeftrightarrow }$	${\displaystyle ~A~}$
	${\displaystyle ~\oplus ~}$		${\displaystyle \Leftrightarrow }$		${\displaystyle \nLeftrightarrow }$

Monotonicity: no

${\displaystyle A\rightarrow B}$	${\displaystyle \nRightarrow }$			${\displaystyle (A\oplus C)}$	${\displaystyle \rightarrow }$	${\displaystyle (B\oplus C)}$
	${\displaystyle \nRightarrow }$		${\displaystyle \Leftrightarrow }$		${\displaystyle \rightarrow }$

Truth-preserving: no

When all inputs are true, the output is not true.

${\displaystyle A\land B}$	${\displaystyle \nRightarrow }$	${\displaystyle A\oplus B}$
	${\displaystyle \nRightarrow }$

Falsehood-preserving: yes

When all inputs are false, the output is false.

${\displaystyle A\oplus B}$	${\displaystyle \Rightarrow }$	${\displaystyle A\lor B}$
	${\displaystyle \Rightarrow }$

Walsh spectrum: (2,0,0,−2)

Non-linearity: 0

The function is linear.

If using binary values for true (1) and false (0), then exclusive or works exactly like addition modulo 2.

It is also called "not left-right arrow" (\nleftrightarrow) in Latex-based markdown (${\displaystyle \nleftrightarrow }$). Apart from the ASCII codes, the operator is encoded at U+22BB ⊻ XOR (&veebar;) and U+2295 ⊕ CIRCLED PLUS (&CirclePlus;, &oplus;), both in block mathematical operators.

Wikimedia Commons has media related to Exclusive disjunction.

Look up exclusive or or XOR in Wiktionary, the free dictionary.

• All About XOR
• Proofs of XOR properties and applications of XOR, CS103: Mathematical Foundations of Computing, Stanford University