The logical operation 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 “trueif and only if exactly one of the operands has a value of “true”. Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence Logical connectiveIn Logic, a set of symbols is commonly used to express logical representation In Mathematics, an operand is one of the inputs (arguments of an Operator. [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. In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence

Contents

Truth table

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

pq$\oplus$
FFF
FTT
TFT
TTF

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. A truth table is a Mathematical table used in Logic — specifically in connection with Boolean algebra, Boolean functions and Propositional

Venn diagram

The Venn diagram of $A \oplus B$ (red 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. Venn diagrams or set diagrams are Diagrams that show all hypothetically possible Logical relations between a finite collection of sets (groups Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence

The exclusive disjunction $p \oplus q$ can be expressed in terms of the logical 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. In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition The proof of this identity is given below:

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

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. In Logic, De Morgan's laws or De Morgan's theorem are rules in Formal logic relating pairs of dual Logical operators in a systematic manner expressed

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. In Logic and Mathematics, logical biconditional (sometimes also known as the material biconditional) is a Logical operator connecting two statements The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain Conditionals in Logic

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 systems $(\{T, F\}, \land)$ and $(\{T, F\}, \lor)$ are monoids. In Mathematics, an operator is a function which operates on (or modifies another function In Logic and/or Mathematics, logical conjunction or and is a two-place Logical operation that results in a value of true if both of In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation This unfortunately prevents the combination of these two systems into larger structures, such as a mathematical ring. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

However, the system using exclusive or $(\{T, F\}, \oplus)$ is an abelian group. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the The combination of operators $\land$ and $\oplus$ over elements {T,F} produce the well-known field F2. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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. English is a West Germanic language originating in England and is the First language for most people in the United Kingdom, the United States 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, any binary or other n-ary exclusive “or” is true if and only if it has an odd number of true inputs. In Logic, Mathematics, and Computer science, the arity (synonyms include type, adicity, and rank) of a function 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:
pqp + q
000
011
101
110
• 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. Addition is the mathematical process of putting things together In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction
• 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. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured C++ (" C Plus Plus " ˌsiːˌplʌsˈplʌs is a general-purpose Programming language. Python is a general-purpose High-level programming language. Its design philosophy emphasizes programmer productivity and code readability This is not used outside of programming contexts because it is too easily confused with other uses of the caret.
• The symbol .
• 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. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, associativity is a property that a Binary operation can have In Mathematics, commutativity is the ability to change the order of something without changing the end result 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

Main article: Bitwise operation

Exclusive disjunction is often used for bitwise operations. The word linear comes from the Latin word linearis, which means created by lines. A logic gate performs a logical operation on one or more logic inputs and produces a single logic output In Computer programming, a bitwise operation operates on one or two Bit patterns or binary numerals at the level of their individual Bits On most 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 $(\Z/2\Z)^n$. In Elementary arithmetic a carry is a Digit that is transferred from one Column of digits to another column of more significant digits during a calculation In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

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 Mathematics, the parity of an object states whether it is even or odd

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. In electronics an adder or summer is a Digital circuit that performs Addition of numbers Symbols There are two symbols for XOR gates the 'military' symbol and the 'rectangular' symbol

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. Traditionally the term neural network had been used to refer to a network or circuit of biological neurons.

Exclusive-or is sometimes used as a simple mixing function in cryptography, for example, with one-time pad or Feistel network systems. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" In Cryptography, the one-time pad (OTP is an Encryption Algorithm where the Plaintext is combined with a random key or "pad" In Cryptography, a Feistel cipher is a symmetric structure used in the construction of Block ciphers named after the German IBM cryptographer Horst

XOR is used in RAID 3–6 for creating parity information. RAID — which stands for Redundant Array of Inexpensive Disks,or alternatively Redundant Array of Independent Disks (a less specific name and thus now the 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 programming, the XOR swap is an Algorithm that uses the XOR Bitwise operation to swap distinct values of Variables

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. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data For code compliance see Bounding. In Computer graphics and Computational geometry, a bounding volume In computing a cursor is an indicator used to show the position on a Computer monitor or other Display device that will respond to input from a text input or In Computer graphics, alpha compositing is the process of combining an image with a background to create the appearance of partial transparency

Notes

1. ^ See Stanford Encyclopedia of Philosophy, article Disjunction
The Logical fallacy of affirming a disjunct also known as the fallacy of the alternative disjunct occurs when a deductive argument takes either of In Boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the inverse of logical or. Boolean algebra (or Boolean logic) is a logical calculus of truth values, developed by George Boole in the late 1830s This is a list of topics around Boolean algebra and propositional logic. In Mathematics and Abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and In Mathematics, a (finitary Boolean function is a function of the form f: B k &rarr B, where B  = {0 1} A boolean-valued function, in some usages a predicate or a proposition, is a function of the type f: X → B, where X is an arbitrary set The Controlled NOT gate (also C-NOT or CNOT) is a Quantum gate that is an essential component in the construction of a Quantum computer. A disjunctive syllogism, historically known as Modus tollendo ponens, is a classically valid, simple Argument form: P or Q First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science A logical graph is a special type of diagramatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true In Logic and Mathematics, the minimal negation operator \nu\! is a Multigrade operator (\nu_{k}_{k \in \mathbb{N}} where each In Logic and Mathematics, a multigrade operator \Omega is a Parametric operator with parameter k in the set In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values In Logic and Mathematics, a parametric operator \Omega\! with parameter \alpha\! in the parametric set \Alpha\! Error detection If an odd number of bits (including the parity bit are changed in transmission of a set of bits then parity bit will be incorrect and will thus indicate This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" In Mathematics, the symmetric difference of two sets is the set of elements which are in one of the sets but not in both XOR linked lists are a Data structure used in Computer programming. Symbols There are two symbols for XOR gates the 'military' symbol and the 'rectangular' symbol In Cryptography, a simple XOR cipher is a relatively simple Encryption algorithm that operates according to the principles A \oplus 0 The Stanford Encyclopedia of Philosophy (SEP is a freely-accessible Online encyclopedia of Philosophy maintained by Stanford University.

exclusive or

-noun

1. (logic, computing) Exclusive disjunction: the use of or to indicate that of two predicates, one is true and one is false (without specifying which is which); contrasted with inclusive or, which does not imply that one must be false.
2. (logic, computing, more generally) Exclusive disjunction: the use of or to indicate that of two or more predicates, an odd number are true (without specifying which or how many); contrasted with inclusive or, which indicates only that one or more is true.
3. (logic, computing) An exclusive disjunction; the result of applying the above-described exclusive or to two or more predicates; contrasted with an inclusive or, which is the result of applying an inclusive or.
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