To be or not to be, that is the proposition

Boolean algebra is a foundation for several fields of mathematics, logic, and computing, including

• propositional calculus
• set theory
• computer hardware design
• computer programming
• document search criteria.

A Boolean algebra is an abstract algebra consisting of

• a set of values (V)
• two dyadic operators (functions from V x V into V):
• 'and', 'conjunction', 'least upper bound', or 'intersection'
• 'or', 'disjunction', 'greatest lower bound', or 'union'
• one monadic operator (function from V into V): 'not', 'negation',  'inverse', or 'complement'
• two distinguished values:
• 'false', 0, or 'the empty set'
• 'true', 1, or 'the universe of discourse'

and which satisfies the following rules for all a, b, and c in V (where ^ is 'and', v is 'or', and ~ is 'not'):

• a v a = a                                             idempotent
• a ^ a = a
• a v b = b v a                                       commutative
• a ^ b = b ^ a
• a v 0 = a                                             identity
• a ^ 1 = a
• a ^ 0 = 0                                             annihilator
• a v 1 = 1
• a ^ ~a = 0                                           complementation
• a v ~a = 1
• a ^ (a v b) = a                                    absorption
• a v (a ^ b) = a
• (a v b) v c = a v (b v c)                      associative
• (a ^ b) ^ c = a ^ (b ^ c)
• a v (b ^ c) = (a v b) ^ (a v c)             distributive
• a ^ (b v c) = (a ^ b) v (a ^ c).

We can define a partial order < where a < b if and only if a v b = b (or, equivalently, a ^ b = a) and a partial order > where a > b if and only if b < a.  The are not total orders because the least upper bound and greatest lower bound for some pairs of values may be some other value.

Additional properties of a Boolean algebra:

• ~(~a) = a                                          double negation
• ~(a v b) = ~a ^ ~b                            DeMorgan's laws
• ~(a ^ b) = ~a v ~b
• Duality:  If we replace 0 with 1, 1 with 0, 'and' with 'or', 'or' with 'and', < with >, and > with < throughout any true statement, the resulting statement is true.  If we make these replacements throughout any false statement, the resulting statement is false.