Advanced Mathematics

2.1Basic Operations of Sets

Purpose of Section

We now present an informal discussion of some of the basic ideas related to sets, including membership in sets, subsets, the empty set, power set, union and intersection of sets, and other fundamental concepts. The material presented in this section will provide a foundation for study in all areas of mathematics.

2.1.1 Sets and Membership

Sets are (arguably) the most fundamental of all mathematical objects. Quite simply, a set is

a collection things, the things belonging to the set are called members

or elements of the set.¹ Other synonyms for the word “set,” are collection, class, family, and ensemble. We refer to a collection of people, family of functions, an ensemble of voters, and so on. We even consider sets whose members themselves are sets, such as the set of classes at a university, where each class consists of students. In mathematics, we might consider the set of all open intervals on the real line or the set of solutions of an equation.

If a set does not contain too many members, we can specify the set by simply writing down the members of the inside a pair of brackets, such as {3, 7, 31}, which denotes the first three Mersenne primes.² Sometimes sets contain an infinite number of elements, like the natural numbers, where we might specify them by {1, 2, 3, …}, where the three dots after the 3 signify “and so on.”

We generally denote sets by capital letters such as A, B, C, … and members of sets by small letters, ...