Boolean Algebra

A Boolean algebra is a structure (B, and, or, not, 0, 1) such that, B is a non-empty set, and and or are binary functions from BxB to B, not is a unary function from B to B, and both 0 and 1 are constants which are members of the set B. The following laws hold:

• for all a in B: a and a = a - idempotency.
• for all a in B: a or a = a - idempotency.
• for all a, b in B: a and b = b and a - commutativity.
• for all a, b in B: a or b = b or a - commutativity.
• for all a, b, c in B: a and (b and c) = (a and b) and c - associativity.
• for all a, b, c in B: a or (b or c) = (a or b) or c - associativity.
• for all a, b, c in B: a and (b or c) = (a and b) or (a and c) - distributivity.
• for all a, b, c in B: a or (b and c) = (a or b) and (a or c) - distributivity.
• for all a in B: a and 1 = a - verum.
• for all a in B: a or 1 = 1 - verum.
• for all a in B: a and 0 = 0 - falsum.
• for all a in B: a or 0 = a - falsum.

Example

We can use digital voltage levels and logic gates as the interpretations of the elements of Boolean algebra as follows:

• B is the set {low voltage level, high voltage level}.
• and is the AND gate.
• or is the OR gate.
• not is the NOT gate.
• 0 is the low voltage level.
• 1 is the high voltage level.

It can be shown by the logic gate truth tables that this interpretation satisfies the laws of Boolean algebra.