Exclusive or or exclusive disjunction is a
logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is
symbolized by the prefix operator J and by the
infix operators XOR ( or ), EOR, EXOR,
⊻,
⩒,
⩛,
⊕,
, and
≢. The
negation of XOR is the
logical biconditional, which yields true if and only if the two 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".
Since it is associative, it may be considered to be an ''n''-ary operator which is true if and only if an odd number of arguments are true. That is, ''a'' XOR ''b'' XOR ... may be treated as XOR(''a'',''b'',...).
Truth table
The
truth table of A XOR B shows that it outputs true whenever the inputs differ:
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
, also denoted by
? or
, can be expressed in terms of the
logical conjunction ("logical and",
), the
disjunction ("logical or",
), and the
negation (
) as follows:
:
The exclusive disjunction
can also be expressed in the following way:
:
This representation of XOR may be found useful when constructing a circuit or network, because it has only one
operation and small number of
and
operations. A proof of this identity is given below:
:
It is sometimes useful to write
in the following way:
:
or:
:
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
Material is a substance or mixture of substances that constitutes an object. Materials can be pure or impure, living or non-living matter. Materials can be classified on the basis of their physical and chemical properties, or on their geologic ...
.
In summary, we have, in mathematical and in engineering notation:
:
Negation
The spirit of De Morgan's laws can be applied, we have:
Relation to modern algebra
Although the
operators (
conjunction) and
(
disjunction) are very useful in logic systems, they fail a more generalizable structure in the following way:
The systems
and
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
''is'' an
abelian group. The combination of operators
and
over elements
produce the well-known
field . This field can represent any logic obtainable with the system
and has the added benefit of the arsenal of algebraic analysis tools for fields.
More specifically, if one associates
with 0 and
with 1, one can interpret the logical "AND" operation as multiplication on
and the "XOR" operation as addition on
:
:
Using this basis to describe a boolean system is referred to as
algebraic normal form
In Boolean algebra, the algebraic normal form (ANF), ring sum normal form (RSNF or RNF), '' Zhegalkin normal form'', or ''Reed–Muller expansion'' is a way of writing logical formulas in one of three subforms:
* The entire formula is purely tru ...
.
Exclusive or in natural language
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.
: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.
: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
Pragmatism is a philosophical movement.
Pragmatism or pragmatic may also refer to:
*Pragmaticism, Charles Sanders Peirce's post-1905 branch of philosophy
* Pragmatics, a subfield of linguistics and semiotics
*'' Pragmatics'', an academic journal i ...
conversational implicature
In pragmatics, a subdiscipline of linguistics, an implicature is something the speaker suggests or implies with an utterance, even though it is not literally expressed. Implicatures can aid in communicating more efficiently than by explicitly sayi ...
s 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.
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''.
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
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
2, which has the following addition table, clearly
isomorphic to the one above:
*
, a modified plus sign; this symbol is also used in mathematics for the ''
direct sum'' of algebraic structures
*
J, as in J''pq''
* An inclusive disjunction symbol (
) that is modified in some way, such as
**
**
*
^, the
caret, used in several
programming languages, such as
C,
C++,
C#,
D,
Java,
Perl,
Ruby,
PHP and
Python, denoting the
bitwise
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operat ...
XOR operator; not used outside of programming contexts because it is too easily confused with other uses of the caret such as exponentiation.
*
, sometimes written as
**
><
**
>-<
*
=1, in IEC symbology
Properties
If using
binary values for true (1) and false (0), then ''exclusive or'' works exactly like
addition modulo
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
2.
Computer science
Bitwise operation
Exclusive disjunction is often used for bitwise operations. Examples:
* 1 XOR 1 = 0
* 1 XOR 0 = 1
* 0 XOR 1 = 1
* 0 XOR 0 = 0
* XOR = (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 .
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
Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric.
Odd may also refer to:
Acronym
* ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...
number of 1 bits (
is true
if and only if an odd number of the variables are true), which is equal to the
parity bit returned by a
parity function.
In logical circuits, a simple
adder can be made with an
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, modeling 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
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or '' -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adv ...
, for example, with
one-time pad or
Feistel network systems.
Exclusive-or is also heavily used in block ciphers such as AES (Rijndael) or Serpent and in block cipher implementation (CBC, CFB, OFB or CTR).
Similarly, XOR can be used in generating
entropy pools for
hardware random number generators. The XOR operation preserves randomness, meaning that a random bit XORed with a non-random bit will result in a random bit. Multiple sources of potentially random data can be combined using XOR, and the unpredictability of the output is guaranteed to be at least as good as the best individual source.
XOR is used in
RAID 3–6 for creating parity information. For example, RAID can "back up" bytes and from two (or more) hard drives by XORing the just mentioned bytes, resulting in () and writing it 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. For instance, if the drive containing is lost, and 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 "1" 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.
XOR linked lists leverage XOR properties in order to save space to represent
doubly linked list data structures.
In
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, XOR-based drawing methods are often used to manage such items as
bounding boxes and
cursors on systems without
alpha channels
In computer graphics, alpha compositing or alpha blending is the process of combining one image with a background to create the appearance of partial or full transparency. It is often useful to render picture elements (pixels) in separate pas ...
or overlay planes.
Encodings
It is also called "not left-right arrow" (
\nleftrightarrow
) in
LaTeX-based markdown (
). Apart from the ASCII codes, the operator is encoded at and , both in block
mathematical operators
Mathematical Operators is a Unicode block containing characters for mathematical, logical, and set notation.
Notably absent are the plus sign (+), greater than sign (>) and less than sign (<), due to them already appearing in the Bas ...
.
See also
Notes
External links
All About XORProofs of XOR properties and applications of XOR, CS103: Mathematical Foundations of Computing, Stanford University
{{Logical connectives
Dichotomies
Logical connectives
Semantics