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Concept
Exclusive or
Summary
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 and by the infix operators XOR (ˌɛks_ˈɔ:r, ˌɛks_ˈɔ:, 'ksɔ:r or 'ksɔ:), EOR, EXOR, , , , ⩛, , and .
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,...).
The truth 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 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 , also denoted by or , can be expressed in terms of the logical conjunction ("logical and", ), the disjunction ("logical or", ), and the negation () 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 applying De Morgan's laws twice to the fourth line of the above proof.
Official source
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