Editing Exclusive or (section)

==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:
* <math>+</math> was used by [[George Boole]] in 1847.<ref name="boole1847">{{cite book |last1=Boole |first1=G. |title=The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning |date=1847 |publisher=Macmillan, Barclay, & Macmillan/George Bell |location=Cambridge/London |page=17 |url=https://archive.org/details/mathematicalanal00booluoft}}</ref> Although Boole used <math>+</math> mainly on classes, he also considered the case that <math>x,y</math> are propositions in <math>x+y</math>, and at the time <math>+</math> is a connective. Furthermore, Boole used it exclusively. Although such use does not show the relationship between inclusive disjunction (for which <math>\vee</math> 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.<ref name="enderton2001">{{cite book |last1=Enderton |first1=H. |title=A Mathematical Introduction to Logic |orig-date=1972 |date=2001 |publisher=A Harcourt Science and Technology Company |location=San Diego, New York, Boston, London, Toronto, Sydney and Tokyo |page=51 |edition=2}}</ref><ref name="rautenberg2010">{{cite book |last1=Rautenberg |first1=W. |title=A Concise Introduction to Mathematical Logic |orig-date=2006 |date=2010 |publisher=Springer |location=New York, Dordrecht, Heidelberg and London |page=3 |edition=3}}</ref>
* <math>\overline{\vee}</math> was used by [[Christine Ladd-Franklin]] in 1883.<ref name="ladd1883">{{cite encyclopedia |last1=Ladd |first1=Christine |title=On the Algebra of Logic |url=https://archive.org/details/studiesinlogic00peiruoft/page/16|encyclopedia=Studies in Logic by Members of the Johns Hopkins University |editor1-last=Peirce |editor1-first=C. S. |publisher=Little, Brown & Company |location=Boston |date=1883 |pages=17–71}}</ref> Strictly speaking, Ladd used <math>A\operatorname{\overline{\vee}}B</math> to express "<math>A</math> is-not <math>B</math>" or "No <math>A</math> is <math>B</math>", i.e., used <math>\overline{\vee}</math> as exclusions, while implicitly <math>\overline{\vee}</math> has the meaning of exclusive disjunction since the article is titled as "On the Algebra of Logic".
* <math>\not=</math>, denoting the negation of [[Logical biconditional|equivalence]], was used by [[Ernst Schröder (mathematician)|Ernst Schröder]] in 1890,<ref name="schroder1890">{{cite book |last1=Schröder |first1=E. |title=Vorlesungen über die Algebra der Logik (Exakte Logik), Erster Band |date=1890 |publisher=Druck und Verlag B. G. Teubner |location=Leipzig |language=German}} Reprinted by Thoemmes Press in 2000.</ref>{{rp|page=307}} Although the usage of <math>=</math> as equivalence could be dated back to [[George Boole]] in 1847,<ref name="boole1847"/> during the 40 years after Boole, his followers, such as [[Charles Sanders Peirce]], [[Hugh MacColl]], [[Giuseppe Peano]] and so on, did not use <math>\not=</math> as non-equivalence literally which is possibly because it could be defined from negation and equivalence easily.
* <math>\circ</math> was used by [[Giuseppe Peano]] in 1894: "<math>a\circ b=a-b\,\cup\,b-a</math>. The sign <math>\circ</math> corresponds to Latin ''aut''; the sign <math>\cup</math> to ''vel''."<ref name="peano1894"> {{cite book |last1=Peano |first1=G. |title=Notations de logique mathématique. Introduction au formulaire de mathématique |date=1894 |publisher=Fratelli Boccna. |location=Turin}} Reprinted in {{cite book |last1=Peano |first1=G. |title=Opere Scelte, Volume II |url=https://archive.org/details/operescelte0002gius/page/n5/mode/2up |date=1958 |publisher=Edizioni Cremonese |location=Roma |pages=123–176}}</ref>{{rp|page=10}} Note that the Latin word "aut" means "exclusive or" and "vel" means "inclusive or", and that Peano use <math>\cup</math> as inclusive disjunction.
* <math>\vee\vee</math> was used by Izrail Solomonovich Gradshtein (Израиль Соломонович Градштейн) in 1936.<ref name="gradshtein1959">{{cite book |last1=ГРАДШТЕЙН |first1=И. С. |title=ПРЯМАЯ И ОБРАТНАЯ ТЕОРЕМЫ: ЭЛЕМЕНТЫ АЛГЕБРЫ ЛОГИКИ |url=https://www.mathedu.ru/text/gradshteyn_pryamaya_i_obratnaya_teoremy_1959/p0/ |orig-date=1936 |date=1959 |publisher=ГОСУДАРСТВЕННОЕ ИЗДАТЕЛЬСТВО ФИЗИКа-МАТЕМАТИЧЕСКОЙ ЛИТЕРАТУРЫ |location=МОСКВА |edition=3 |language=Russian}} Translated as {{cite book |last1=Gradshtein |first1=I. S. |translator-last1=Boddington |translator-first1=T. |title=Direct and Converse Theorems: The Elements of Symbolic Logic |date=1963 |publisher=Pergamon Press |location=Oxford, London, New York and Paris}}</ref>{{rp|page=76}}
* <math>\oplus</math> was used by [[Claude Shannon]] in 1938.<ref>{{ cite journal
|last = Shannon
|first = C. E.
|author-link = Claude Elwood Shannon
|title = A Symbolic Analysis of Relay and Switching Circuits
|journal = Transactions of the American Institute of Electrical Engineers
|year = 1938
|volume = 57 |issue=12
|pages = 713–723
|doi= 10.1109/T-AIEE.1938.5057767
|hdl = 1721.1/11173
|s2cid = 51638483
|url = https://www.cs.virginia.edu/~evans/greatworks/shannon38.pdf
|hdl-access = free
}}</ref> Shannon borrowed the symbol as exclusive disjunction from [[Edward Vermilye Huntington]] in 1904.<ref name="huntington1904">{{cite journal |last1=Huntington |first1=E. V. |title=Sets of Independent Postulates for the Algebra of Logic |journal=Transactions of the American Mathematical Society |date=1904 |volume=5 |issue=3 |pages=288–309|doi=10.1090/S0002-9947-1904-1500675-4 }}</ref> Huntington borrowed the symbol from [[Gottfried 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).<ref>{{cite book |last1=Leibniz |first1=G. W. |editor1-last=Gerhardt |editor1-first=C. I. |title=Die philosophischen Schriften, Siebter Band |url=https://archive.org/details/diephilosophisc01leibgoog/page/n11/mode/2up |access-date= 7 July 2023 |orig-date=16??/17??|date=1890 |language= German |publisher=Weidmann |location=Berlin |page=237}}</ref> 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 <math>+</math> as inclusive disjunction.<ref name="huntington1933">{{cite journal |last1=Huntington |first1=E. V. |title=New Sets of Independent Postulates for the Algebra of Logic, With Special Reference to Whitehead and Russell's Principia Mathematica |journal=Transactions of the American Mathematical Society |date=1933 |volume=35 |issue=1 |pages=274–304}}</ref>
* <math>\not\equiv</math>, also denoting the negation of [[Logical biconditional|equivalence]], was used by [[Alonzo Church]] in 1944.<ref name="church1944">{{cite book |last1=Church |first1=A. |title=Introduction to Mathematical Logic |orig-date=1944|date=1996 |publisher=Princeton University Press |location=New Jersey |page=37}}</ref>
* <math>J</math> (as a [[Polish notation|prefix operator]], <math>J\phi\psi</math>) was used by [[Józef Maria Bocheński]] in 1949.<ref name="Bochenski1949"/>{{rp|page=16}} Somebody<ref name="Craig_1998">{{cite book |title=Routledge Encyclopedia of Philosophy, Volume 8 |author-first=Edward |author-last=Craig |publisher=[[Taylor & Francis]] |date=1998 |isbn=978-0-41507310-3 |page=496 |url=https://books.google.com/books?id=mxpFwcAplaAC&pg=PA496}}</ref> may mistake that it is [[Jan Łukasiewicz]] who is the first to use <math>J</math> for exclusive disjunction (it seems that the mistake spreads widely), while neither in 1929<ref name="lukasiewicz1929">{{cite book |title=Elementy logiki matematycznej |language=pl |trans-title=Elements of Mathematical Logic |author-last=Łukasiewicz |author-first=Jan |author-link=Jan Łukasiewicz |location=Warsaw, Poland |edition=1 |publisher=[[Państwowe Wydawnictwo Naukowe]] |date=1929}}</ref> nor in other works did Łukasiewicz make such use. In fact, in 1949 Bocheński introduced a system of [[Polish notation]] that names all 16 binary [[logical connective|connectives]] of classical logic which is a compatible extension of the notation of Łukasiewicz in 1929, and in which <math>J</math> for exclusive disjunction appeared at the first time. Bocheński's usage of <math>J</math> 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.
* <samp>^</samp>, the [[caret]], has been used in several [[programming language]]s to denote the [[bitwise operation|bitwise]] exclusive or operator, beginning with [[C (programming language)|C]]<ref>{{cite book|title=The C Programming Language|title-link=The C Programming Language|year=1978|publisher=Prentice-Hall|first1=Brian W.|last1=Kernighan|author1-link=Brian Kernighan|first2=Dennis M.|last2=Ritchie|author2-link=Dennis Ritchie|contribution=2.9: Bitwise logical operators|pages=44–46|contribution-url=https://archive.org/details/TheCProgrammingLanguageFirstEdition/page/n51}}</ref> and also including [[C++]], [[C Sharp (programming language)|C#]], [[D (programming language)|D]], [[Java (programming language)|Java]], [[Perl]], [[Ruby (programming language)|Ruby]], [[PHP]] and [[Python (programming language)|Python]].
* The [[symmetric difference]] of two sets <math>S</math> and <math>T</math>, which may be interpreted as their elementwise exclusive or, has variously been denoted as <math>S\ominus T</math>, <math>S\mathop{\triangledown} T</math>, or <math>S\mathop{\vartriangle} T</math>.<ref>{{mathworld|title=Symmetric Difference|urlname=SymmetricDifference}}</ref>