Set Algebra

Abstract

The set of positive integers is ℕ; the set of real numbers is ℝ. The set of all subsets of a set X is 2^X. If E ⊂ 2^X, then R(E), (σR(E), A(E), σA(E)) is the intersection of the (nonempty) set of rings (σ- rings, algebras, σ- algebras) containing E and contained in 2^X. It is the ring (σ-ring, algebra, σ-algebra) generated by E. The set of x in X such that···is {x:···}. If A ⊂ X then A’ = {x : x ∉ A} and if B ⊂ X then A\B = A ∩ B’. The cardinality of X is card(X). If X is a topological space then O(X) (F(X), K(X)) is the set of open (closed, compact) subsets of X. A subset M of 2^X is monotone if it is closed with respect to the formation of countable unions and intersections of monotone sequences in M, i.e., if {A _n : n = 1, 2,...} is a sequence in M and A _n ⊂ A _{n +1} (A _n ⊃ A _{n +1}) for n in ℕ then ⋃ _n A _n (∩ _n A_n) is in M. The set M(E) is the monotone subset of 2 ^X generated by E.