# Set Theory

• Houshang H. Sohrab

## Abstract

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 sS, and its negation will be denoted sS. 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 ST, if each element of S is an element of T. Again the negation of the statement will be denoted ST. One obviously has ∅ ⊂ S for any set S. We write S = T if both ST and TS. S is called a proper subset of T if ST, but ST. In this case one also says that the inclusion ST is a proper inclusion. We shall constantly use the notation S = {tT : 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.

## Keywords

Equivalence Relation Commutative Ring Nonempty Subset Identity Element Choice Function