The logical operation exclusive disjunction, also called exclusive or (symbolized by the prefix operator J, or by the infix operators XOR, EOR, EXOR, or , play /ˌɛks ˈɔr/ or /ˈzɔr/), is a type of logical disjunction on two operands that results in a value of true if exactly one of the operands has a value of true.[1] A simple way to state this is "one or the other but not both."

Put differently, exclusive disjunction is a logical operation on two logical values, typically the values of two propositions, that produces a value of true only in cases where the truth value of the operands differ.

The opposite of XOR is logical biconditional, where the output of two compared values is true only if both A and B are the same.

Exclusive disjunction essentially means 'either one, but not both'. In other words, if and only if one is true, the other cannot be true. For example, one of the two horses will win the race, but not both of them. The exclusive disjunction :p oplus q, or Jpq, can be expressed in terms of the logical conjunction (wedge), the disjunction (lor), and the negation (lnot) as follows:

p oplus q & = & (p land lnot q) lor (lnot p land q)

The exclusive disjunction p oplus q can also be expressed in the following way:

p oplus q & = & lnot (p land q) land (p lor q)

This representation of XOR may be found useful when constructing a circuit or network, because it has only one lnot operation and small number of wedge and lor operations. The proof of this identity is given below:

p oplus q & = & (p land lnot q) & lor & (lnot p land q) \
& = & ((p land lnot q) lor lnot p) & and & ((p land lnot q) lor q) \
& = & ((p lor lnot p) land (lnot q lor lnot p)) & land & ((p lor q) land (lnot q lor q)) \
& = & (lnot p lor lnot q) & land & (p lor q) \
& = & lnot (p land q) & land & (p lor q)

It is sometimes useful to write p oplus q in the following way:

p oplus q & = & lnot ((p land q) lor (lnot p land lnot q))

This equivalence can be established by applying De Morgan's laws twice to the fourth line of the above proof.

The exclusive or is also equivalent to the negation of a logical biconditional, by the rules of material implication (a material conditional is equivalent to the disjunction of the negation of its antecedent and its consequence) and material equivalence.

In summary, we have, in mathematical and in engineering notation:

p oplus q & = & (p land lnot q) & lor & (lnot p land q) & = & poverline{q} + overline{p}q \
      & = & (p lor q) & land & (lnot p lor lnot q) & = & (p+q)(overline{p}+overline{q}) \
      & = & (p lor q) & land & lnot (p land q) & = & (p+q)(overline{pq})

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"The waiver program has been absolutely the most spectacular thing for people with dementia, especially in rural areas, because there aren't the day cares, there aren't the other resources for them," said Sara Willis, a social worker with Tennova's Geriatric Assessment Program (GAP). "It's huge, and our families are using it. They have the respite they need to at least keep their head above water and cope.

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The agency heads also received instructions from Linda J. Yelmini, the director of labor relations at the governor's budget office and one of the most knowledgeable state officials on labor issues. Yelmini has been working on the ongoing talks with the unions that have been headed by chief negotiator Mark Ojakian, a longtime aide to then state comptroller Nancy Wyman.

When you go to park your bike, there nothing to attach it to, she said. I think if businesses can put a small bike rack out front, it really encouraging. People can bring their bikes and they feel more comfortable knowing it safe leaving it out front.

The students were offered a prize either a small check for the next day or a larger check at some later date.The men made perfectly rational decisions, opting for the delayed larger amount after viewing the average looking women. You know where this is going. (Women, by the way, were rational no matter what.) That men are at a disadvantage when attractive women are present is a fact upon which women have banked for centuries.
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