A Boolean algebra is a set A, supplied with two binary operations ∧ (called AND), ∨ (called OR), a unary operation ¬ (called NOT) and two elements 0 (called zero) and 1 (called one), such that, for all elements a, b and c of set A, the following axioms hold:
The first three pairs of axioms above: associativity, commutativity and absorption, mean that (A, ∧, ∨) is a lattice. If A is a lattice and one of the above distributivity laws holds, then the second distributivity law can be proven. Thus, a Boolean algebra can also be equivalently defined as a distributive complemented lattice.
From these axioms, one can show that the smallest element 0, the largest element 1, and the complement ¬a of any element a are uniquely determined. For all a and b in A, the following identities also follow:
[These axioms have been extracted from Wikipedia]
To show that the complement of an element is unique, assume that x and y are both complements of a. Then
x ∧ (y ∨ a) = x ∧ 1 = x
But also:
x ∧ (y ∨ a) = (x ∧ y) ∨ (x ∧ a) = (x ∧ y) ∨ 0 = x ∧ y
By symmetry, it follows that
x = x ∧ y = y ∧ x = y