Exclusive or

In logical operations, the exclusive disjunction, also called exclusive or, (symbolized XOR or EOR), is a type of logical disjunction on two operands that results in a value of “true” if and only if exactly one of the operands has a value of “true”. ^[1]

Put differently, exclusive disjunction is a logical operation on two logical values, typically the values of two propositions, that produces a value of true just in cases where the truth value of the operands differ.

Truth table

The truth table of $p\,\mathrm {XOR} \,q$ (also written as $p\oplus q$ , or $p\neq q$ ) is as follows:


$p$	$q$	$\oplus$
$T$	$T$	$F$
$T$	$F$	$T$
$F$	$T$	$T$
$F$	$F$	$F$

Note the three-way symmetry of the outcomes: The identity of $p$ , $q$ , and $\neq$ in this table could be arbitrarily re-assigned, and the table would still be correct.

Venn diagram

The Venn diagram of $A\oplus B$ (white part is true)

Equivalencies, elimination, and introduction

The following equivalents can then be deduced, written with logical operators, in mathematical and engineering notation:

{\begin{matrix}p\oplus q&=&(p\land \lnot q)&\lor &(\lnot p\land q)=p{\overline {q}}+{\overline {p}}q\\\\&=&(p\lor q)&\land &(\lnot p\lor \lnot q)=(p+q)({\overline {p}}+{\overline {q}})\\\\&=&(p\lor q)&\land &\lnot (p\land q)=(p+q)({\overline {pq}})\end{matrix}}

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

The exclusive disjunction $p\oplus q$ can be expressed in terms of the conjunction ( $\land$ ), the disjunction ( $\lor$ ), and the negation ( $\lnot$ ) as follows:

{\begin{matrix}p\oplus q&=&(p\land \lnot q)\lor (\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&=&\lnot (p\land q)\land (p\lor 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 $\land$ and $\lor$ operations. The proof of this identity is given below:

{\begin{matrix}p\oplus q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\&=&((p\land \lnot q)\lor \lnot p)&\land &((p\land \lnot q)\lor q)\\&=&((p\lor \lnot p)\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}}

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}}

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 a the disjunction of the negation of its antecedent and its consequence) and material equivalence.

Relation to modern algebra

Although the operators $\land$ (conjunction) and $\lor$ (disjunction) are very useful in logic systems, the latter fails a more generalizable structure in the following way:

The system $(\{T,F\},\land )$ is an abelian group but the system $(\{T,F\},\lor )$ is a monoid. 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 $\land$ 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.

Exclusive “or” in natural language

The Oxford English Dictionary explains “either … or” as follows:

The primary function of either, etc., is to emphasize the indifference of the two (or more) things or courses … but a secondary function is to emphasize the mutual exclusiveness, = either of the two, but not both.

Following this kind of common-sense intuition about “or”, it is sometimes argued that in many natural languages, English included, the word “or” has an “exclusive” sense. The exclusive disjunction of a pair of propositions, (p, q), is supposed to mean that p is true or q is true, but not both. For example, it is argued, the normal intention of a statement like “You may have coffee or you may have tea” is to stipulate that exactly one of the conditions can be true. Certainly under many circumstances a sentence like this example should be taken as forbidding the possibility of one's accepting both options. Even so, there is good reason to suppose that this sort of sentence is not disjunctive at all. If all we know about some disjunction is that it is true overall, we cannot be sure that either of its disjuncts is true. For example, if a woman has been told that her friend is either at the snack bar or on the tennis court, she cannot validly infer that he is on the tennis court. But if her waiter tells her that she may have coffee or she may have tea, she can validly infer that she may have tea. Nothing classically thought of as a disjunction has this property. This is so even given that she might reasonably take her waiter as having denied her the possibility of having both coffee and tea.

There are also good general reasons to suppose that no word in any natural language could be adequately represented by the binary exclusive “or” of formal logic. First, the binary exclusive “or” is true if and only if it has an odd number of true inputs. But it seems as though no word in any natural language that can conjoin a list of two or more options has this general property. Second, as pointed out by Barrett and Stenner in the 1971 article “The Myth of the Exclusive ‘Or’” (Mind, 80 (317), 116–121), no author has produced an example of an English or-sentence that appears to be false because both of its inputs are true. Certainly there are many or-sentences such as “The light bulb is either on or off” in which it is obvious that both disjuncts cannot be true. But it is not obvious that this is due to the nature of the word “or” rather than to particular facts about the world.

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 ( $+$ ). This makes sense mathematically because exclusive disjunction corresponds to addition modulo 2, which has the following addition table, clearly isomorphic to the one above:

**Addition Modulo 2**
$p$	$q$	$p+q$
0	0	0
0	1	1
1	0	1
1	1	0

The use of the plus sign has the added advantage that all of the ordinary algebraic properties of mathematical rings and fields can be used without further ado. However, the plus sign is also used for Inclusive disjunction in some notation systems.
A plus sign that is modified in some way, such as being encircled ( $\oplus$ ). 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 ( $\lor$ ) that is modified in some way, such as being underlined ( ${\underline {\lor }}$ ) or with dot above ( ${\dot {\vee }}$ ).
In several programming languages, such as C, C++, Python and Java, a caret (^) is used to denote the bitwise XOR operator. This is not used outside of programming contexts because it is too easily confused with other uses of the caret.
The symbol File:X or.svg.
In IEC symbology, an exclusive or is marked “=1”.

Properties

This section uses the following symbols:

{\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}}

The following equations follow from logical axioms:

{\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}}

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:

{\begin{matrix}p+q&=&q+p\\\\(p+q)+r&=&p+(q+r)&=&p+q+r\end{matrix}}

Other properties

falsehood preserving: The interpretation under which all variables are assigned a truth value of ‘false’ produces a truth value of ‘false’ as a result of exclusive disjunction.
linear

Computer science

Traditional symbolic representation of an XOR logic gate

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 $(\mathbb {Z} /2\mathbb {Z} )^{n}$ .

In computer science, exclusive disjunction has several uses:

It tells whether two bits are unequal.
It is an optional bit-flipper (the deciding input chooses whether to invert the data input).
It tells whether there is an odd number of 1 bits ( $A\oplus B\oplus C\oplus D\oplus E$ is true iff an odd number of the variables are true).

In logical circuits, a simple adder can be made with a XOR gate to add the numbers, and a series of AND, OR and NOT gates to create the carry output.

On some computer architectures, it is more 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 networks, 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 RAID 3–6 for creating parity information. For example, RAID can “back up” bytes 10011100 and 01101100 from two (or more) hard drives by XORing (11110000) and writing to another drive. Under this method, if any one of the three hard drives are lost, the lost byte can be re-created by XORing bytes from the remaining drives. If the drive containing 01101100 is lost, 10011100 and 11110000 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 swap two numeric variables in computers, using the XOR swap algorithm; however this is regarded as more of a curiosity and not encouraged in practice.

In computer graphics, XOR-based drawing methods are often used to manage such items as bounding boxes and cursors on systems without alpha channels or overlay planes.

Notes

^ See Stanford Encyclopedia of Philosophy, article Disjunction

Template:Logical Operators

[1] See Stanford Encyclopedia of Philosophy, article Disjunction

[1]