Appendix

Summary

Boolean algebras

In this appendix, X is always a (nonempty) set and S is a nonempty set of subsets of X.For any A⊆X, the symbol A^⊥ denotes the set theoretical complement of A in X;thatis, A^⊥ = X\A.

Definition A.1S is a (Boolean) ring if for every A, B ϵ S we have (A∪B) ϵ S and (A\B) ϵ S.

A ring is a set of sets that is closed with respect to set theoretical union and difference. If S is a ring, then ∅ ϵ S (because ∅ = A\A); however, X does not necessarily belong to S. If it does, then the (Boolean) ring is called Boolean algebra:

Definition A.2 A Boolean ring S is called Booelan algebra if X ϵ S.

A Boolean algebra S is thus closed with respect to the complement: if S is a Boolean algebra and AϵS, then A^⊥ ϵ S.

One can define the notion of Boolean algebra directly: S is a Boolean algebra with respect to the set theoretical operations ∪, ∩, ⊥ if X ϵ S and if it holds that if A, B ϵ S, then (A∩B), (A∪B), and A^⊥ are all in S.