Boolean algebra
A Boolean algebra is the algebra of ‘true and false’, or equivalently of ‘sets and their complements’. It captures the laws that the logical connectives AND, OR, NOT obey, and equally the laws that intersection, union and complement of subsets obey — these turn out to be exactly the same laws. It is the mathematical heart of propositional logic, digital circuits and the algebra of events in probability.
Formally, a Boolean algebra is a complemented distributive lattice: a bounded distributive lattice (L, ∧, ∨, 0, 1) in which every element a has a complement ¬a satisfying a ∧ ¬a = 0 and a ∨ ¬a = 1. Equivalently, it is an algebra with operations (∧, ∨, ¬, 0, 1) satisfying the Boolean identities (including De Morgan's laws ¬(a ∧ b) = ¬a ∨ ¬b). In a distributive lattice complements, when they exist, are unique, so ¬ is a genuine operation and Boolean algebras form a variety.
Stone's representation theorem gives the definitive picture: every Boolean algebra is isomorphic to a field of sets — a subalgebra of some power set under ∩, ∪, complement — and indeed to the algebra of clopen sets of a compact totally disconnected (Stone) space. The two-element Boolean algebra {0, 1} is the building block: an identity holds in all Boolean algebras iff it holds in {0, 1}, which is why truth tables decide propositional validity.
The power set of {1, 2, 3} with ∩, ∪ and complement (relative to {1, 2, 3}) is a Boolean algebra of 8 elements; here 0 = ∅, 1 = {1, 2, 3}, and ¬{1} = {2, 3}.
A finite Boolean algebra always has 2^n elements and is a power set up to isomorphism.