The purpose of this section is to define a boolean algebra and to determine some of the important properties of it.
A boolean algebra is a set B with two binary operations, join and meet, denoted by + and juxtaposition respectively, and a unary operation, complement ation, denoted by \u27, which satisfy the following axioms:
(1) for all a,b ∑ B (that is, for all a,b elements of B) a + b = b + a and a b = b a, (the commutative laws),
(2) for all a,b,c ∑ B, a + b c =(a + b) (a + b) and a (b + c) = a b + a c, (the distributive laws),
(3) there exists 0 ∑B such that for each a ∑B, a + 0 = a, and there exists 1 ∑B such that for each a ∑ B, a 1 = a,
(4) for each a∑B, a + a\u27 = 1 and a a\u27 = 0.
If a + e = a for all a in B then 0 = 0 + e = e + 0 = e, so that there is exactly one element in B which satisfies the first half of axiom 3, namely 0. Similarly there is exactly one element in B which satisfies the second half of axiom 3, namely 1.
The O and 1 as defined above will be called the distinguished elements
