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exclusive or
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Truth table
File:Multigrade operator XOR.svg|thumb|220px|Arguments on the left combined by XOR. This is a binary Walsh matrix (cf. Hadamard codeHadamard codeThe truth table of A XOR B shows that it outputs true whenever the inputs differ:{| class="wikitable" style="text-align:center"|+ XOR truth table!colspan="2" | Input || rowspan="2" | Output!A || B| 0| 1| 1| 0- 0, false
- 1, true
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 p oplus q, also denoted by p â©› q or operatorname{J}pq, can be expressed in terms of the logical conjunction ("logical and", wedge), the disjunction ("logical or", lor), and the negation (lnot) as follows:
begin{matrix}
p oplus q & = & (p lor q) land 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 & = & (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 lnot operation and small number of wedge and lor operations. A proof of this identity is given below:
begin{matrix}
p oplus 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 p oplus q in the following way:
& = & ((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)
begin{matrix}
p oplus q & = & lnot ((p land q) lor (lnot p land lnot q))
end{matrix}or:
begin{matrix}
p oplus 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:
begin{matrix}
p oplus q & = & (p land lnot q) & lor & (lnot p land q) & = & poverline{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}& = & (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})
Relation to modern algebra
Although the operators wedge (conjunction) and lor (disjunction) are very useful in logic systems, they fail a more generalizable structure in the following way:The systems ({T, F}, wedge) and ({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 ({T, F}, oplus) is an abelian group. The combination of operators wedge 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.More specifically, if one associates F with 0 and T with 1, one can interpret the logical "AND" operation as multiplication on F_2 and the "XOR" operation as addition on F_2:
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}Using this basis to describe a boolean system is referred to as algebraic normal form.r = p oplus q & Leftrightarrow & r = p + q pmod 2
Exclusive "or" in English
{{essay|section|date=May 2013}}The Oxford English Dictionary explains "either ... or" as follows:
either of the two, but not both.or, conj.2 (adv.3) 2a Oxford English Dictionary, second edition (1989). OED Online.}}
The exclusive-or explicitly states "one or the other, but not neither nor both." However, the mapping correspondence between formal Boolean operators and natural language conjunctions is far from simple or one-to-one, and has been studied for decades in linguistics and analytic philosophy.{{citation needed|date=January 2013}}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.Jennings quotes numerous authors saying that the word "or" has an exclusive sense. See Chapter 3, "The First Myth of 'Or'":BOOK, Jennings, R. E., 1994, The Genealogy of Disjunction, New York, Oxford University Press, 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 might be argued that 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 some 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 which 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.{{citation needed|date=June 2012}}In English, the construct "either ... or" is usually used to indicate exclusive or and "or" generally used for inclusive.{{dubious|date=January 2013}} But in Spanish, the word "o" (or) can be used in the form "p o q" (exclusive) or the form "o p o q" (inclusive). Some may contend that any binary or other n-ary exclusive "or" is true if and only if it has an odd number of true inputs (this is not, however, the only reasonable definition; for example, digital xor gates with multiple inputs typically do not use that definition), and that there is no conjunction in English that has this general property. For example, Barrett and Stenner contend in the 1971 article "The Myth of the Exclusive 'Or{{' "}} (Mind, 80 (317), 116â€“121) that no author has produced an example of an English or-sentence that appears to be false because both of its inputs are true, and brush off or-sentences such as "The light bulb is either on or off" as reflecting particular facts about the world rather than the nature of the word "or". However, the "barber paradox"â€”Everybody in town shaves himself or is shaved by the barber, who shaves the barber? -- would not be paradoxical if "or" could not be exclusive (although a purist could say that "either" is required in the statement of the paradox).Whether these examples can be considered "natural language" is another question.{{dubious|date=January 2013}} Certainly when one sees a menu stating "Lunch special: sandwich and soup or salad" (parsed as "sandwich and (soup or salad)" according to common usage in the restaurant trade), one would not expect to be permitted to order both soup and salad. Nor would one expect to order neither soup nor salad, because that belies the nature of the "special", that ordering the two items together is cheaper than ordering them a la carte. Similarly, a lunch special consisting of one meat, French fries or mashed potatoes and vegetable would consist of three items, only one of which would be a form of potato. If one wanted to have meat and both kinds of potatoes, one would ask if it were possible to substitute a second order of potatoes for the vegetable. And, one would not expect to be permitted to have both types of potato and vegetable, because the result would be a vegetable plate rather than a meat plate.{{citation needed|date=June 2012}}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 2, which has the following addition table, clearly isomorphic to the one above:
- oplus, a modified plus sign; this symbol is also used in mathematics for the direct sum of algebraic structures
- J, as in Jpq
- An inclusive disjunction symbol (lor) that is modified in some way, such as
- underlinelor
- dotvee
- ^, the caret, used in several programming languages, such as C, C++, C, D, Java, Perl, Ruby, PHP and Python, denoting the bitwise XOR operator; not used outside of programming contexts because it is too easily confused with other uses of the caret
- (File:X-or.svg|24px), sometimes written as
- '''>-
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