Exclusive or,exclusive disjunction,exclusive alternation,logical non-equivalence, orlogical inequality is alogical operator whose negation is thelogical biconditional. With two inputs, XOR is true if and only if the inputs differ (one is true, one is false). With multiple inputs, XOR is true if and only if the number of true inputs isodd.[1]
It gains the name "exclusive or" because the meaning of "or" is ambiguous when bothoperands are true. XORexcludes that case. Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B".
Each row of this binaryWalsh matrix is the truth table of thevariadic XOR of the arguments shown on the left.E.g. row AB corresponds to the 2-circle, and row ABC to the 3-circle Venn diagram shown above. (As in the Venn diagrams, white is false, and red is true.)
Thetruth table of shows that it outputs true whenever the inputs differ:
Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is trueif 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 thelogical conjunction ("logical and",), thedisjunction ("logical or",), and thenegation () 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 applyingDe Morgan's laws twice to the fourth line of the above proof.
Although theoperators (conjunction) and (disjunction) are very useful in logic systems, they fail a more generalizable structure in the following way:
The systems and aremonoids, but neither is agroup. This unfortunately prevents the combination of these two systems into larger structures, such as amathematical ring.
However, the system using exclusive oris anabelian group. The combination of operators and over elements produce the well-knowntwo-element 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:
Disjunction is often understood exclusively innatural 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.[4][5]
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 befelicitously used in combination with an outright statement that both disjuncts are true. The second example shows that the exclusive inference vanishes away underdownward 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.[4]
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 aspragmaticconversational implicatures calculated on the basis of an inclusivesemantics. Implicatures are typicallycancellable and do not arise in downward entailing contexts if their calculation depends on theMaxim of Quantity. However, some researchers have treated exclusivity as a bona fide semanticentailment and proposed nonclassical logics which would validate it.[4]
This behavior of English "or" is also found in other languages. However, many languages have disjunctive constructions which are robustly exclusive such as Frenchsoit... soit.[4]
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:
was used byGeorge Boole in 1847.[6] Although Boole used mainly on classes, he also considered the case that are propositions in, and at the time is a connective. Furthermore, Boole used it exclusively. Although such use does not show the relationship between inclusive disjunction (for which is almost fixedly used nowadays) and exclusive disjunction, and may also bring about confusions with its other uses, some classical and modern textbooks still keep such use.[7][8]
was used byChristine Ladd-Franklin in 1883.[9] Strictly speaking, Ladd used to express " is-not" or "No is", i.e., used as exclusions, while implicitly has the meaning of exclusive disjunction since the article is titled as "On the Algebra of Logic".
, denoting the negation ofequivalence, was used byErnst Schröder in 1890,[10]: 307 Although the usage of as equivalence could be dated back toGeorge Boole in 1847,[6] during the 40 years after Boole, his followers, such asCharles Sanders Peirce,Hugh MacColl,Giuseppe Peano and so on, did not use as non-equivalence literally which is possibly because it could be defined from negation and equivalence easily.
was used byGiuseppe Peano in 1894: ". The sign corresponds to Latinaut; the sign tovel."[11]: 10 Note that the Latin word "aut" means "exclusive or" and "vel" means "inclusive or", and that Peano use as inclusive disjunction.
was used by Izrail Solomonovich Gradshtein (Израиль Соломонович Градштейн) in 1936.[12]: 76
was used byClaude Shannon in 1938.[13] Shannon borrowed the symbol as exclusive disjunction fromEdward Vermilye Huntington in 1904.[14] Huntington borrowed the symbol fromGottfried Wilhelm Leibniz in 1890 (the original date is not definitely known, but almost certainly it is written after 1685; and 1890 is the publishing time).[15] While both Huntington in 1904 and Leibniz in 1890 used the symbol as an algebraic operation. Furthermore, Huntington in 1904 used the symbol as inclusive disjunction (logical sum) too, and in 1933 used as inclusive disjunction.[16]
(as aprefix operator,) was used byJózef Maria Bocheński in 1949.[2]: 16 Somebody[18] may mistake that it isJan Łukasiewicz who is the first to use for exclusive disjunction (it seems that the mistake spreads widely), while neither in 1929[19] nor in other works did Łukasiewicz make such use. In fact, in 1949 Bocheński introduced a system ofPolish notation that names all 16 binaryconnectives of classical logic which is a compatible extension of the notation of Łukasiewicz in 1929, and in which for exclusive disjunction appeared at the first time. Bocheński's usage of as exclusive disjunction has no relationship with the Polish "alternatywa rozłączna" of "exclusive or" and is an accident for which see the table on page 16 of the book in 1949.
The exclusive or does not distribute over any binary function (not even itself), butlogical conjunction distributes over exclusive or. (Conjunction and exclusive or form the multiplication and addition operations of afieldGF(2), and as in any field they obey the distributive law.)
Exclusive or with one specified input, as a function of the other input, is aninvolution or self-inverse function; applying it twice leaves the variable input unchanged.
If usingbinary values for true (1) and false (0), thenexclusive or works exactly likeadditionmodulo 2.
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
11102 XOR10012 =01112 (this is equivalent to addition withoutcarry)
As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of twon-bit strings is identical to the standard vector of addition in thevector space.
In computer science, exclusive disjunction has several uses:
It tells whether two bits are unequal.
It is a controllable bit-flipper (the control input chooses whether or not to invert the data input).
It tells whether there is anodd number of 1 bits ( is trueif and only if an odd number of the variables are true), which is equal to theparity bit returned by aparity function.
In logical circuits, a simpleadder can be made with anXOR 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) than to load and store the value zero.
Incryptography, XOR is sometimes used as a simple, self-inverse mixing function, such as inone-time pad orFeistel network systems.[citation needed] XOR 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 generatingentropy pools forhardware 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.[22]
XOR is used inRAID 3–6 for creating parity information. For example, RAID can "back up" bytes100111002 and011011002 from two (or more) hard drives by XORing the just mentioned bytes, resulting in (111100002) 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 containing011011002 is lost,100111002 and111100002 can be XORed to recover the lost byte.[23]
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 theXOR swap algorithm; however this is regarded as more of a curiosity and not encouraged in practice.
It is also called "not left-right arrow" (\nleftrightarrow) inLaTeX-based markdown (). Apart from the ASCII codes, the operator is encoded atU+22BB⊻XOR (⊻) andU+2295⊕CIRCLED PLUS (⊕, ⊕), both in blockmathematical operators.
^abcdAloni, Maria (2016)."Disjunction". In Zalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy (Winter 2016 ed.). Metaphysics Research Lab, Stanford University. Retrieved2020-09-03.
^Jennings quotes numerous authors saying that the word "or" has an exclusive sense. See Chapter 3, "The First Myth of 'Or'": Jennings, R. E. (1994).The Genealogy of Disjunction. New York: Oxford University Press.
^Enderton, H. (2001) [1972].A Mathematical Introduction to Logic (2 ed.). San Diego, New York, Boston, London, Toronto, Sydney and Tokyo: A Harcourt Science and Technology Company. p. 51.
^Rautenberg, W. (2010) [2006].A Concise Introduction to Mathematical Logic (3 ed.). New York, Dordrecht, Heidelberg and London: Springer. p. 3.
^Ladd, Christine (1883)."On the Algebra of Logic". In Peirce, C. S. (ed.).Studies in Logic by Members of the Johns Hopkins University. Boston: Little, Brown & Company. pp. 17–71.
^Schröder, E. (1890).Vorlesungen über die Algebra der Logik (Exakte Logik), Erster Band (in German). Leipzig: Druck und Verlag B. G. Teubner. Reprinted by Thoemmes Press in 2000.
^Peano, G. (1894).Notations de logique mathématique. Introduction au formulaire de mathématique. Turin: Fratelli Boccna. Reprinted inPeano, G. (1958).Opere Scelte, Volume II. Roma: Edizioni Cremonese. pp. 123–176.
^ГРАДШТЕЙН, И. С. (1959) [1936].ПРЯМАЯ И ОБРАТНАЯ ТЕОРЕМЫ: ЭЛЕМЕНТЫ АЛГЕБРЫ ЛОГИКИ (in Russian) (3 ed.). МОСКВА: ГОСУДАРСТВЕННОЕ ИЗДАТЕЛЬСТВО ФИЗИКа-МАТЕМАТИЧЕСКОЙ ЛИТЕРАТУРЫ. Translated asGradshtein, I. S. (1963).Direct and Converse Theorems: The Elements of Symbolic Logic. Translated by Boddington, T. Oxford, London, New York and Paris: Pergamon Press.
^Huntington, E. V. (1904). "Sets of Independent Postulates for the Algebra of Logic".Transactions of the American Mathematical Society.5 (3):288–309.doi:10.1090/S0002-9947-1904-1500675-4.
^Huntington, E. V. (1933). "New Sets of Independent Postulates for the Algebra of Logic, With Special Reference to Whitehead and Russell's Principia Mathematica".Transactions of the American Mathematical Society.35 (1):274–304.
^Church, A. (1996) [1944].Introduction to Mathematical Logic. New Jersey: Princeton University Press. p. 37.
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