Introduction to Relations

Published: May 15th, 2020

Learning Objectives
By the end of this lesson, you should be able to:
1. Define the term "relation"
2. Describe the domain and range of a relation
3. Describe the ways by which a relation is represented.

We use functions to describe relationships where one quantity depends on another. For example, the amount of time it takes you to get somewhere depends on how fast you are travelling. 

This section will discuss the terminology of functions. To succeed in this course, you should be well acquainted with these terms. To better understand how these terms relate, we will use an example. A farmer plants various crops. Interested in mathematics, he observes that his revenue for a week scales with the number of crops he plants.  He notes that the number of crops he sells and the revenue he earns in a week are related, and drew a graph to represent this relation.

A relation is a rule that associates each element (denoted by \(x\), typically) from some set with element(s) (denoted as \(y\), typically) from another set.

In this case, the farmer is relating his crops planted (an independent variable, \(x\)) with the revenue (the dependent variable, \(y\)). Although we're representing this relation in graph form, there are many other ways we can represent relations. For example, with a table of values, as shown to the left. We can also represent this relation as a set of ordered pairs: \((1, 20), (2, 40), (3, 60), and \:(4, 80)\)

The set of all x values for which a relation exists is called its domain, while the set of all y values is called the range. We represent these values using set notation \(\left\{\right\}\) (curly brackets). For this relation, the domain is: \(\left\{ 1, 2, 3, 4 \right\}\) and the range is \(\left\{20,40,60,80\right\}\)

A relation can also be expressed through an equation such as the equations \(y=mx+b\) (linear relationship) or \(y=x^2\) (quadratic relationship), which you may have seen before. 

The diagram below summarizes the different ways we can represent relations, and provides a summary of this lesson.

 

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