Exclusive or

Contra- is in classical logic and mathematics, the name for the connection of two statements by the two- connective, either - is, or, exclusive or, and also Kontravalentor.

Synonymous with contra- be used the designations exclusive disjunction (also complete or non-equivalent disjunction ), Bisubtraktion, or exclusionary, non-equivalence, contradictory contrast, Kontrajunktion age or nation. In the switching algebra is called the XOR gate, in propositional logic it is called XOR operation.

Definition and properties

Defines the contra- is determined by the truth values ​​of a function of their connective either which is a contra- is only true if both of them associated statements have different truth values ​​, that is, when one or the other true if there are not both true or both at the same time wrong. The Latin term for this exclusive Or as " either - or" is " aut - aut ".

By means of a truth table (matrix) the aut - function is defined as a function of the truth values ​​contra- thus given as follows:

The contra- is associative and commutative. In addition, it is self-inverse and distributive with respect to logical AND, but not for OR:

Demarcation and similarities

The difference to the non- exclusive-or (in the narrow sense, the disjunction ) is the "enhanced information " that " from the outset it is clear that one of the two alternatives must be true ", that is not more than one of only at least, but the two situations there.

Equivalences of the contra-, ie formulas with other connectives that have the same truth value course are:

• Negation of Bikonditionals (negation of equivalence )
• Or
• Or
• .

Meaning and practical application

The importance of contra- 's modern logic rather low, " since it allows to formulate relatively few contexts ". In the Boolean algebra has as XORing the other hand, great importance. The property that the repeated application of the XOR operation corresponds to the identity, that is, that it is self-inverse is, among other things, in cryptography - used and the RAID system - there she allows to use the same function when encrypting and decrypting. See also application of the XOR operation.

Notation and pronunciation

Symbols of Kontravalentors include:

• ⌴ half upwardly open square.
• XOR
• "> - <"
• "> < "

The speech of the connective also varies:

• "A contrasting B"
• " A or (but ) B '
• "Either A or B"
• " A, except that B"
• " A except that B"
• "A, except that B "
• "A iff not B"

Colloquially the Kontravalentor with is " either - or" paraphrased. But this expression can colloquially also have other meanings that do not comply with the contra-; For example, can " pick you up either Emil or I from " even be understood as true if both pick up the interlocutor.