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Encyclopedia > Exclusive or

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 to what extent a proposition is true. ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ...

Truth table

The truth table of $p, mathrm{XOR}, q$ (also written as $p oplus q$, or $p neq q$) is as follows: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

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) A Venn diagram of sets A, B, and C Venn diagrams are illustrations used in the branch of mathematics known as set theory. ...

Image File history File links No higher resolution available. ...

The following equivalents can then be deduced, written with logical operators, in mathematical and engineering notation: In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ...

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

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:

$begin{matrix} p + q & = & (p land lnot q) lor (lnot p land q) end{matrix}$

The exclusive disjunction $p oplus q$ can also be expressed in the following way:

$begin{matrix} p + q & = & lnot (p land q) land (p lor q) end{matrix}$

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:

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

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. note that demorgans laws are also a big part in circut design. ...

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 logical calculus of mathematics, logical biconditional is a logical operator connecting two statements to assert, p if and only if q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain conditionals in logic. ... â†” â‡” â‰¡ logical symbols representing iff. ...

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: OR logic gate. ...

• 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 F2. 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. In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

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. The English language is a West Germanic language that originates in England. ...

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, n-ary 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:
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 ($underlinelor$) or with dot above ($dotvee$).
• 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 .
• In IEC symbology, an exclusive or is marked “=1”.

3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, the direct sum is a construction which combines several modules into a new, bigger one. ... A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... C is a general-purpose, block structured, procedural, imperative computer programming language developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories for use with the Unix operating system. ... C++ (pronounced see plus plus, IPA: ) is a general-purpose programming language with high-level and low-level capabilities. ... Python is a high-level programming language first released by Guido van Rossum in 1991. ... â€œJava languageâ€ redirects here. ... Image File history File links This is a lossless scalable vector image. ...

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: In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, associativity is a property that a binary operation can have. ... Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation). ...

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

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

For other uses, see Linear (disambiguation). ...

Computer science

Traditional symbolic representation of an XOR logic gate

Image File history File links Xor-gate-en. ... Image File history File links Xor-gate-en. ... A logic gate performs a logical operation on one or more logic inputs and produces a single logic output. ...

Bitwise operation

Main article: Bitwise operation

Exclusive disjunction is often used for bitwise operations. Examples: In computer programming, a bitwise operation operates on one or two bit patterns or binary numerals at the level of their individual bits. ...

• 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 $(Z/2Z)^n$. 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 algorithm. ... 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).

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 computer science, efficiency is used to describe several desirable properties of an algorithm or other construct, besides clean design, functionality, etc. ...

In simple threshold activated neural networks, modelling 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 circuitry 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. The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages Cryptography (or cryptology; derived from Greek ÎºÏÏ…Ï€Ï„ÏŒÏ‚ kryptÃ³s hidden, and the verb Î³ÏÎ¬Ï†Ï‰ grÃ¡fo write or Î»ÎµÎ³ÎµÎ¹Î½ legein to speak) is the study of message secrecy. ... Excerpt from a one-time pad. ... In cryptography, a Feistel cipher is a block cipher with a symmetric structure, named after IBM cryptographer Horst Feistel; it is also commonly known as a Feistel network. ...

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. For other uses, see Raid. ...

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. It has been suggested that this article or section be merged with swap (computer science). ...

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. This article is about the scientific discipline of computer graphics. ... A bounding box for a three dimensional model For code compliance, see Bounding. ... A blinking text cursor. ... In computer graphics, alpha compositing is often useful to render image elements in separate passes, and then combine the resulting multiple 2D images into a single, final image in a process called compositing. ...

Notes

1. ^ See Stanford Encyclopedia of Philosophy, article Disjunction

Results from FactBites:

 Exclusive or - Wikipedia, the free encyclopedia (1119 words) Exclusive or (usual symbol XOR occasionally EOR), which is sometimes called exclusive disjunction, is a logical operator that results in true if one of the operands, but not both of them, is true. The exclusive disjunction of propositions A and B is usually called A xor B, where "xor" stands for "exclusive or" and is pronounced "eks-or" or "zor". Exclusive-or is sometimes used as a simple mixing function in cryptography, for example, with one-time pad or Feistel network systems.
 Talk:Exclusive or - Wikipedia, the free encyclopedia (775 words) "Exclusive disjunction is the sense of the word or as in the proverb you can have your cake or eat it (but not both)." which is not supported by the definition of exclusive disjunction. In other words, if the meaning in the proverb were really the exclusive disjunction, it would follow that I had permission to take one of the two actions, but you wouldn't tell me which.
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