A **set** is a group of objects, numbers, and so forth. {1,2,3} is a set consisting of the numbers 1,2, and 3. Verbally, “3 is an element of the set {1,2,3}.” To show this symbolically, use the symbol ∈, which is read as “is an element of” or “is a member of.” Therefore, you could have written:

3 ∈ {1,2,3}

Special sets

A **subset** is a set contained within another set, or it can be the entire set itself. The set {1,2} is a subset of the set {1,2,3}, and the set {1,2,3} is a subset of the set {1,2,3}. When the subset is missing some elements that are in the set it is being compared to, it is a **proper subset.** When the subset is the set itself, it is an **improper subset.** The symbol used to indicate “is a proper subset of” is ⊂. When there is the possibility of using an improper subset, the symbol used is ⊆. Therefore, {1,2} ⊂ {1,2,3} and {1,2,3} ⊆ {1,2,3}. The **universal set** is the general category set, or the set of all those elements under consideration. The **empty set,** or **null set,** is the set with no elements or *members*. The empty set, or null set, is represented by ⊘, or { }. However, it is never represented by {⊘}.

Both the universal set and the empty set are subsets of every set.

Describing sets

**Rule** is a method of naming a set by describing its elements.

{ *x*: *x* > 3, *x* is a whole number} describes the set with elements 4, 5, 6,…. Therefore, { *x*: *x* > 3, *x* is a whole number} is the same as {4,5,6,…}. { *x*: *x* > 3} describes all numbers greater than 3. This set of numbers cannot be represented as a list and is represented using a number line graph.

**Roster** is a method of naming a set by listing its members.

{1,2,3} is the set consisting of only the elements 1,2, and 3. There are many ways to represent this set using a rule. Two correct methods are as follows:

{ *x*: *x* < 4, *x* is a natural number}

{ *x*:0 < *x* < 4, *x* is a whole number}

An incorrect method would be { *x*:0 < *x* < 4} because this rule includes ALL numbers between 0 and 4, not just the numbers 1, 2, and 3.

Types of sets

**Finite sets** have a countable number of elements. For example, { *a,b,c,d,e*} is a set of five elements, thus it is a finite set. **Infinite sets** contain an uncountable number of elements. For example, {1,2,3, …} is a set with an infinite number of elements, thus it is an infinite set.

Comparing sets

**Equal sets** are those that have the exact same members — {1, 2, 3} = {3, 2, 1}. **Equivalent sets** are sets that have the same number of members — {1, 2, 3} | { *a, b, c*}.

**Venn diagrams** (and *Euler circles*) are ways of pictorially describing sets as shown in Figure 1.

Figure 1. A Venn diagram

The *A* represents all the elements in the smaller oval; the *B* represents all the elements in the larger oval; and the *C* represents all the elements that are in both ovals at the same time.

Operations with sets

The **union** of two sets is a set containing all the numbers in those sets, but any duplicates are only written once. The symbol for finding the union of two sets is ∪.

Example 1

Find the union {1,2,3} ∪ {3,4,5}.

{1,2,3} ∪ {3,4,5} = {1,2,3,4,5}

The union of the set with members 1, 2, 3 together with the set with members 3, 4, 5 is the set with members 1, 2, 3, 4, 5.

The **intersection** of two sets is a set containing only the members that are in each set at the same time. The symbol for finding the intersection of two sets is ∩.

Example 2

Find the intersection {1,2,3} ∩ {3,4,5}.

{1,2,3} ∩ {3,4,5} = {3}

The intersection of the set with members 1, 2, 3 together with the set with members 3, 4, 5 is the set that has only the 3.

If you were to let the set with {1,2,3} be set *A*, and the set with {3,4,5} be set *B*, then you could use Venn diagrams to illustrate the situation (see Figure 2).

Figure 2. Intersection of set A and set B

The union will be all the numbers represented in the diagram, {1,2,3,4,5}.The intersection would be where the two ovals overlap in the diagram, {3}.

Example 3

Find {1,2,3} ∩ {4,5}.

Since there are no members that are in both sets at the same time, then {1,2,3} ∩ {4,5} = ⊘.

The intersection of the set with members 1, 2, 3 together with the set with members 4, 5 is the empty set, or null set. There are no members in both sets at the same time.