Exclusive or

Each row of this binary Walsh matrix is the truth table of the variadic XOR of the arguments shown on the left. E.g. row AB corresponds to the 2-circle, and row ABC to the 3-circle Venn diagram shown above. (As in the Venn diagrams, white is false, and red is true.)

The truth table of ${\displaystyle A\oplus B}$ 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 ${\displaystyle p\operatorname {?} q}$ or ${\displaystyle Jpq}$, 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:

$${\displaystyle \lnot (p\nleftrightarrow q)\Leftrightarrow \lnot p\nleftrightarrow q\Leftrightarrow p\nleftrightarrow \lnot q.}$$

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 two-element field ${\displaystyle \mathbb {F} _{2}}$. 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 ${\displaystyle \mathbb {F} _{2}}$ and the "XOR" operation as addition on ${\displaystyle \mathbb {F} _{2}}$:

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

The description of a Boolean function as a polynomial in ${\displaystyle \mathbb {F} _{2}}$, using this basis, is called the function's algebraic normal form.[3]

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.[4][5]

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.[4]

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.[4]

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.[4]

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:

• ${\displaystyle +}$ was used by George Boole in 1847.[6] Although Boole used ${\displaystyle +}$ mainly on classes, he also considered the case that ${\displaystyle x,y}$ are propositions in ${\displaystyle x+y}$, and at the time ${\displaystyle +}$ is a connective. Furthermore, Boole used it exclusively. Although such use doesn't show the relationship between inclusive disjunction (for which ${\displaystyle \vee }$ is almost fixedly used nowadays) and exclusive disjunction, and may also bring about confusions with its other uses, some classical and modern textbooks still keep such use.[7][8]
• ${\displaystyle {\overline {\vee }}}$ was used by Christine Ladd-Franklin in 1883.[9] Strictly speaking, Ladd used ${\displaystyle A\operatorname {\overline {\vee }} B}$ to express "${\displaystyle A}$ is-not ${\displaystyle B}$" or "No ${\displaystyle A}$ is ${\displaystyle B}$", i.e., used ${\displaystyle {\overline {\vee }}}$ as exclusions, while implicitly ${\displaystyle {\overline {\vee }}}$ has the meaning of exclusive disjunction since the article is titled as "On the Algebra of Logic".
• ${\displaystyle \not =}$, denoting the negation of equivalence, was used by Ernst Schröder in 1890,[10]^: 307 Although the usage of ${\displaystyle =}$ as equivalence could be dated back to George Boole in 1847,[6] during the 40 years after Boole, his followers, such as Charles Sanders Peirce, Hugh MacColl, Giuseppe Peano and so on, didn't use ${\displaystyle \not =}$ as non-equivalence literally which is possibly because it could be defined from negation and equivalence easily.
• ${\displaystyle \circ }$ was used by Giuseppe Peano in 1894: "${\displaystyle a\circ b=a-b\,\cup \,b-a}$. The sign ${\displaystyle \circ }$ corresponds to Latin aut; the sign ${\displaystyle \cup }$ to vel."[11]^: 10 Note that the Latin word "aut" means "exclusive or" and "vel" means "inclusive or", and that Peano use ${\displaystyle \cup }$ as inclusive disjunction.
• ${\displaystyle \vee \vee }$ was used by Izrail Solomonovich Gradshtein (Израиль Соломонович Градштейн) in 1936.[12]^: 76
• ${\displaystyle \oplus }$ was used by Claude Shannon in 1938.[13] Shannon borrowed the symbol as exclusive disjunction from Edward Vermilye Huntington in 1904.[14] Huntington borrowed the symbol from Gottfried Wilhelm Leibniz in 1890 (the original date is not definitely known, but almost certainly it's written after 1685; and 1890 is the publishing time).[15] While both Huntington in 1904 and Leibniz in 1890 used the symbol as an algebraic operation. Furthermore, Huntington in 1904 used the symbol as inclusive disjunction (logical sum) too, and in 1933 used ${\displaystyle +}$ as inclusive disjunction.[16]
• ${\displaystyle \not \equiv }$, also denoting the negation of equivalence, was used by Alonzo Church in 1944.[17]
• ${\displaystyle J}$ (as a prefix operator, ${\displaystyle J\phi \psi }$) was used by Józef Maria Bocheński in 1949.[2]^: 16 Somebody[18] may mistake that it's Jan Łukasiewicz who is the first to use ${\displaystyle J}$ for exclusive disjunction (it seems that the mistake spreads widely), while neither in 1929[19] nor in other works did Łukasiewicz make such use. In fact, in 1949 Bocheński introduced a system of Polish notation that names all 16 binary connectives of classical logic which is a compatible extension of the notation of Łukasiewicz in 1929, and in which ${\displaystyle J}$ for exclusive disjunction appeared at the first time. Bocheński's usage of ${\displaystyle J}$ as exclusive disjunction has no relationship with the Polish "alternatywa rozłączna" of "exclusive or" and is an accident for which see the table on page 16 of the book in 1949.
• ^, the caret, has been used in several programming languages to denote the bitwise exclusive or operator, beginning with C[20] and also including C++, C#, D, Java, Perl, Ruby, PHP and Python.
• The symmetric difference of two sets ${\displaystyle S}$ and ${\displaystyle T}$, which may be interpreted as their elementwise exclusive or, has variously been denoted as ${\displaystyle S\ominus T}$, ${\displaystyle S\mathop {\triangledown } T}$, or ${\displaystyle S\mathop {\vartriangle } T}$.[21]

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 (⊻) and U+2295 ⊕ CIRCLED PLUS (⊕, ⊕), 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