Boolean algebras

Abstract

Let X be an arbitrary non-empty set and let P (X) (the power set of X) be the class of all subsets of X. There is a way of introducing a Boolean structure into P (X), as follows. The distinguished elements are defined by

$$ 0 = \emptyset \:\operatorname{and} \:1 = X, $$

and, if P and Q are subsets of X, then, by definition,

$$ P + Q\left( {P \cap Q'} \right) \cup \left( {P' \cap Q} \right)\;and\;PQ = P \cap Q. $$