Exclusive or

${\displaystyle +}$ was used by in 1847.^[6] Although Boole used ${\displaystyle +}$ mainly on classes, he also considered the case that ${\displaystyle x,y}$ are propositions in ${\displaystyle x+y}$, and at the time ${\displaystyle +}$ is a connective. Furthermore, Boole used it exclusively. Although such use does not show the relationship between inclusive disjunction (for which ${\displaystyle \vee }$ 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]

George Boole

${\displaystyle {\overline {\vee }}}$ was used by in 1883.^[9] Strictly speaking, Ladd used ${\displaystyle A\operatorname {\overline {\vee }} B}$ to express "${\displaystyle A}$ is-not ${\displaystyle B}$" or "No ${\displaystyle A}$ is ${\displaystyle B}$", i.e., used ${\displaystyle {\overline {\vee }}}$ as exclusions, while implicitly ${\displaystyle {\overline {\vee }}}$ has the meaning of exclusive disjunction since the article is titled as "On the Algebra of Logic".

Christine Ladd-Franklin

${\displaystyle \not =}$, denoting the negation of , was used by Ernst Schröder in 1890,^[10]: 307 Although the usage of ${\displaystyle =}$ as equivalence could be dated back to George Boole in 1847,^[6] during the 40 years after Boole, his followers, such as Charles Sanders Peirce, Hugh MacColl, Giuseppe Peano and so on, did not use ${\displaystyle \not =}$ as non-equivalence literally which is possibly because it could be defined from negation and equivalence easily.

equivalence

${\displaystyle \circ }$ was used by in 1894: "${\displaystyle a\circ b=a-b\,\cup \,b-a}$. The sign ${\displaystyle \circ }$ corresponds to Latin aut; the sign ${\displaystyle \cup }$ to vel."^[11]: 10 Note that the Latin word "aut" means "exclusive or" and "vel" means "inclusive or", and that Peano use ${\displaystyle \cup }$ as inclusive disjunction.

Giuseppe Peano

${\displaystyle \vee \vee }$ was used by Izrail Solomonovich Gradshtein (Израиль Соломонович Градштейн) in 1936.^: 76

[12]

${\displaystyle \oplus }$ was used by in 1938.^[13] Shannon borrowed the symbol as exclusive disjunction from Edward Vermilye Huntington in 1904.^[14] 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).^[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 ${\displaystyle +}$ as inclusive disjunction.^[16]

Claude Shannon

${\displaystyle \not \equiv }$, also denoting the negation of , was used by Alonzo Church in 1944.^[17]

equivalence

${\displaystyle J}$ (as a , ${\displaystyle J\phi \psi }$) was used by Józef Maria Bocheński in 1949.^[2]: 16 Somebody^[18] may mistake that it is Jan Łukasiewicz who is the first to use ${\displaystyle J}$ 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 of Polish notation that names all 16 binary connectives of classical logic which is a compatible extension of the notation of Łukasiewicz in 1929, and in which ${\displaystyle J}$ for exclusive disjunction appeared at the first time. Bocheński's usage of ${\displaystyle J}$ 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.

prefix operator

^, the , has been used in several programming languages to denote the bitwise exclusive or operator, beginning with C^[20] and also including C++, C#, D, Java, Perl, Ruby, PHP and Python.

caret

The of two sets ${\displaystyle S}$ and ${\displaystyle T}$, which may be interpreted as their elementwise exclusive or, has variously been denoted as ${\displaystyle S\ominus T}$, ${\displaystyle S\mathop {\triangledown } T}$, or ${\displaystyle S\mathop {\vartriangle } T}$.^[21]

symmetric difference

1 XOR 1 = 0

1 XOR 0 = 1

0 XOR 1 = 1

0 XOR 0 = 0

11102 XOR 10012 = 01112 (this is equivalent to addition without )

carry

All About XOR

Proofs of XOR properties and applications of XOR, CS103: Mathematical Foundations of Computing, Stanford University