# Relation in Math: Definition & Examples

In this lesson, you will learn the definition of relation in terms of mathematics, as well as the various ways of displaying relations. We will also look at some examples.

What is a Relation?

A relation is a relationship between sets of values. In math, the relation is between the x-values and y-values of ordered pairs. The set of all x-values is called the domain, and the set of all y-values is called the range.

In this image, we can see that the domain consists of the x-values from each ordered pair. The range consists of the y-values from each ordered pair. The brackets are used to show that the values form a set. It is important to not repeat values in the domain and range sets (notice that two of the ordered pairs have the number 2 as the y-value, but 2 is listed only once in the range). In this example, the values in the domain and range are listed numerically. While this is common practice, it is not essential.

Domain and range for a set of ordered pairs.

relation1

Displaying Relations

Relations can be displayed as a table, a mapping or a graph. In a table the x-values and y-values are listed in separate columns. Each row represents an ordered pair:

Displaying a relation as a table.

relation2

A mapping shows the domain and range as separate clusters of values. Lines are drawn to match each value in the domain with its corresponding value in the range:

Displaying a relation as a mapping.

relation3

Graphs can also be used to show the relationships between values. Each ordered pair is plotted as a point on the graph. The placement of a point along the x- and y-axes indicate the x- and y-values for the ordered pair:

Displaying a relation as a graph.

relation4

Examples of Relation Problems

In our first example, our task is to create a list of ordered pairs from the set of domain and range values provided.

Domain and range for Example 1.

relation5

At first glance, we may try to pair each value in the order that they are listed, such as (3, -11), (8, -7), etc. However, we cannot assume that the values in each set are ordered so that the pairs match up. Furthermore, there are four values in the domain and five in the range.

It turns out that there is no way for us to complete this problem. We need more information to tell us which values in the domain should be paired with each value in the range. For this reason, it is important to have a table, mapping or graph available. Each of those displays provide a way for us to list the correct ordered pairs. In the next example, we’ll see how a mapping is used to create a list of ordered pairs.

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