exclusive

Exclusive or or exclusive disjunction or exclusive alternation, also known as non-equivalence which is the negation of equivalence, is a logical operation that is true if and only if its arguments differ (one is true, the other is false).It is symbolized by the prefix operator



J


{\displaystyle J}
: 16  and by the infix operators XOR (, , or ), EOR, EXOR,






∨
˙





{\displaystyle {\dot {\vee }}}
,





∨
¯




{\displaystyle {\overline {\vee }}}
,





∨
_




{\displaystyle {\underline {\vee }}}
, ⩛,



⊕


{\displaystyle \oplus }
,



↮


{\displaystyle \nleftrightarrow }
and



≢


{\displaystyle \not \equiv }
.
It gains the name "exclusive or" because the meaning of "or" is ambiguous when both operands are true; the exclusive or operator excludes that case. This is sometimes thought of as "one or the other but not both" or "either one or the other". This could be written as "A or B, but not, A and B".
XOR is equivalent to logical inequality (NEQ) since it is true only when the inputs are different (one is true, and one is false). The negation of XOR is the logical biconditional, which yields true if and only if the two inputs are the same, which is equivalent to logical equality (EQ).
Since it is associative, it may be considered to be an n-ary operator which is true if and only if an odd number of arguments are true. That is, a XOR b XOR ... may be treated as XOR(a,b,...).

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