A set¹ is a collection of objects, called elements of the set. If x is an element of the set A, then we write $x\in A$; otherwise, we write $x\notin A$. For example, if z is the set of integers, then $3\in Z$ and ${\displaystyle \frac{1}{2}}\notin Z$.

One set that appears frequently is the set of real numbers, which we denote by R throughout this text.

Two sets A and B are called equal, written $A=B$, if they contain exactly the same elements. Sets may be described in one of two ways:

1. By listing the elements of the set between set braces $\left\{\right\}$.

2. By describing the elements of the set in terms of some characteristic property.

For example, the set consisting of the elements 1, 2, 3, and 4 can be written as $\{1,2,3,4\}$ or as

Note that the order in which the elements of a set are listed is immaterial, and listing an element in a set more than once does not change the set. Hence

A set B is called a subset of a set A, written $B\subseteq A$ or $A\supseteq B$, if every element of B is an element of A. For example, $\{1,2,6\}\subseteq \{2,8,7,6,1\}$. If $B\subseteq A$, and $B\ne A$, then B is called a proper subset of A. Observe that $A=B$ if and only if $A\subseteq B$ and $B\subseteq A$, a fact that is often used to prove that two sets are equal.

The empty set, denoted by $\varnothing$, is the set containing no elements. The empty set is a subset of every set.

Sets may be combined to form other sets in two basic ways. The union of two sets A and B, denoted $A\cup B$, is the set of elements that are in A, or B, or both; that is,

The intersection of two sets A and B, denoted $A\cap B$, is the set of elements that are in both A and B; that is,

Two sets are called disjoint if their intersection equals the empty set.

The union and intersection of more than two sets can be defined analogously. Specifically, if $A_1,A_2,\dots ,A_n$ are sets, then the union and intersections of these sets are defined, respectively, by

and

In this situation, we have one set $A_i$ for each $i\in \{1,2,\dots ,n\}$. It can be convenient to regard these sets $A_i$ as being indexed by the set $I=\{1,2,\dots ,n\}$. When doing so, I is called an index set for the collection $\{A_i:i\in I\}$.

The use of an index set is especially useful with an infinite collection of sets, as in Example 3 below. If $\Lambda$ is an index set and $\{A_{\alpha }:\alpha \in \Lambda \}$ is a collection of sets, the union and intersection of these sets are defined, respectively, by

and

By a relation on a set A, we mean a rule for determining whether or not, for any elements x and y in A, x stands in a given relationship to y. To make this concept precise, we define a relation on A to be any set S of ordered pairs of elements of A, and say that the elements x and y of A satisfy the relation S if and only if $(x,y)\in S$. On the set of real numbers, for instance, “is equal to,” “is less than,” and “is greater than or equal to” are familiar relations. If S is a relation on a set A, we often write $x\sim y$ in place of $(x,y)\in S$.

A relation S on a set A is called an equivalence relation on A if the following three conditions hold:

1. For each $x\in A,x\sim x$ (reflexivity).

2. If $x\sim y$, then $y\sim x$ (symmetry).

3. If $x\sim y$ and $y\sim z$, then $x\sim z$ (transitivity).

For example, if we define $x\sim y$ to mean that $x\sim y$ is divisible by a fixed integer n, then $\sim$ is an equivalence relation on the set of integers.