Set Theory

A set, S, will be defined as a “collection” (or “family”) of “objects” called elements. The statement “s is an element of S” will be denoted s ∈ S, and its negation will be denoted s ∉ S. The set with no elements will be called the empty set, and denoted ∅. Given a pair of sets, S and T, we say that S is a subset of T, and write S ⊂ T, if each element of S is an element of T. Again the negation of the statement will be denoted S ⊄ T. One obviously has ∅ ⊂ S for any set S. We write S = T if both S ⊂ T and T ⊂ S. S is called a proper subset of T if S ⊂ T, but S ≠ T. In this case one also says that the inclusion S ⊂ T is a proper inclusion. We shall constantly use the notation S = {t ∈ T : P(t)} to denote the set of all elements in T for which the property P holds. In most problems, all the sets we consider are subsets of a fixed (large) set, called the universal set or the universe of discourse, which we denote by U. We will usually assume that such a universe has been chosen, especially when complements of sets (to be defined below) are involved in the discussion. Before defining the basic operations on sets, let us introduce a notation which will be used throughout the book.