# MandysNotes

Tuesday, 18 March 2014 18:31

## Xor: Exclusive Or

By  Gideon
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If all we know is that A is not equivalent to B, we can write this as:

$\lnot \left( A \leftrightarrow B \right)$

$= \lnot \left[ \left( \left( \lnot A \right) \cup B \right) \cap \left( \left( \lnot B \right) \cup A \right) \right]$

We call the set

$= \lnot \left[ \left( \left( \lnot A \right) \cup B \right) \cap \left( \left( \lnot B \right) \cup A \right) \right]$

"A xor B"

where xor means "exclusive or."

An animal that is a member of the set (A xor B) could be a duckling (or any other non-mammal duck-billed animal) or it could be a rabbit (or any other non duck-billed mammal) but it cannot be a platypus, nor can it be any animal that is neither duck-billed nor a mammal.

Xor becomes  a very important relation when we make the transition from Boolean algebras to Boolean rings: xor  is the "addition" relation (conjuntion is the "multiplication" relation).