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1. Relations
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Chapter 8
1. 

Relations

A relation in mathematics shows how elements from one set are associated with elements of another set. Using mapping diagrams helps visualize these relationships, with the domain representing all possible inputs and the range representing possible outputs. Understanding and analyzing these relationships is key for solving problems in various areas of math.

Student Learning Objectives:
  • Identify and represent a relation using a variety of methods
  • Identify the domain and range of a relation
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11 Theory slides
10 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
Relations
Slide of 11
It is common to want to make a connection between two different sets of information. For example, consider counting people at a park. The number of people changes as the temperatures changes. Such relations are used to describe a connection between the elements of two sets. This lesson will dive deep into relations and how to represent them.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Challenge

How to Represent a Relation?

Tadeo, the younger brother of Vincenzo, loves watching Vincenzo's college basketball games. He recorded Vincenzo and his teammates' names, heights, and the number of 3-point shots they made. Tadeo describes the relation between Vincenzo's teammates' height and the number of 3-points shots they made. He is so excited to share them with his big sibling.

Tiffaniqua, 1.70 m, made two 3-point shots; Jordan,1.75 m, made five 3-points shots; Ramsha, 1.85 m, made four 3-point shots; Vincenzo, 1.96 m, made eight 3-point shots; Diego, 1.91 m, made eight 3-point shots

Which of the following diagrams can Tadeo use to illustrate the relation?

Diagrams
Discussion

Relation

A relation, or relationship, is a rule that relates the elements of one set to the elements of another set. The first set is called the set of inputs and the second set is called the set of outputs.

Set of inputs, set of outputs, and an arrow from one set to the other

Often, a relation is thought of as a set of ordered pairs of the form (x,y). In this case, the x-values represent the inputs and the y-values the outputs. However, a relation can also be represented using a mapping diagram, a table of values, or a set of points on a coordinate plane.

relation represented as a mapping diagram, a set of ordered pairs, a table of values, and as points in the coordinate plane
Discussion

Mapping Diagram

A mapping diagram is a graphic tool that helps to visualize a relation. In a mapping diagram, the inputs are listed in one set and the outputs in another. Then, arrows are drawn from each input to its corresponding outputs.

A mapping diagram representing the relation {(-2,-8),(-1,11),(0,5),(2,-8)} from the input set [-2,-1,0,2] to the output set [-8,11,5].
This mapping diagram shows the relation defined by the set {(-2,-8), (-1,11), (0,5), (2,-8)}. Note that -8 is the output of two different inputs, -2 and 2.
Example

Representing a Relation in Different Ways

Tadeo enjoyed sharing his last finding so much that he now wants to examine the relationship between the height and age of Vincenzo's teammates. He wants to describe some relations based on the values in the following diagram.

Tiffaniqua, 16 yo, 1.70 m; Jordan,18 yo, 1.75 m; Ramsha,21 yo, 1.85 m, ; Vincenzo, 22 yo, 1.96 m; Diego, 21 yo, 1.91 m

Represent the relation between Vincenzo's teammates' ages and heights using a table of values, a mapping diagram, a set of ordered pairs, and points in a coordinate plane.

Answer
Table of Values:

Ages 16 18 21 22 21
Heights (m) 1.70 1.75 1.85 1.96 1.91

Mapping Diagram:

Mapping diagram

Ordered Pairs: {(16,1.70), (18,1.75), (21,1.85), (22,1.96), (21,1.91)}
Coordinate Plane:

Points in the coordinate plane: (16,1.70),(18,1.75),(21,1.85),(22,1.96),(21,1.91)

Hint
To make a mapping diagram, place all the ages in one set and all the heights in another set. Then, connect each age with its corresponding height using an arrow. To write the relation as a set of ordered pairs, place the ages in the first component and the corresponding heights in the second component.

Solution
Using the information that Tadeo gathered, a relation between the ages and heights can be made and represented using different visualizations.

Table of Values

First, organize the relation using a table of values. Place the ages in the first row and the heights in the second row. Align the ages with the corresponding heights.

Ages 16 18 21 22 21
Heights (m) 1.70 1.75 1.85 1.96 1.91

Mapping Diagram

The same relation can be displayed using a mapping diagram. Place the ages in a set and the heights in a different set. Then, connect each age with its corresponding height by using arrows. It is not necessary to write 21 twice in the ages set.

Mapping diagram

Set of Ordered Pairs

To write the relation using ordered pairs, the ages will be placed in the first coordinate of each ordered pair, and the corresponding heights will be placed in the second coordinate. { c (16,1.70),(18,1.75),(21,1.85), (22,1.96),(21,1.91) }

Points in the Coordinate Plane

Finally, the relation between the ages and heights of the players can be represented using points in the coordinate plane. To do so, the ages will be placed along the horizontal axis and the heights along the vertical axis. Here, the axes will be drawn using different scales.

Points in the coordinate plane: (16,1.70),(18,1.75),(21,1.85),(22,1.96),(21,1.91)

Note that by hovering over each point, the corresponding information is shown.

Discussion

Domain and Range of a Relation

The domain of a relation is the set of all inputs, for which the relation is defined. For example, consider the following relation. R = {(0,- 2), (1,0), (2,2) } The domain of the relation is the set of all first coordinate of the ordered pairs.

Relation Inputs Domain
{( 0,- 2), ( 1,0), ( 2,2) } 0, 1, 2 { 0, 1, 2 }

Depending on how a relation is represented, its domain can be determined using different methods.

Graph in the coordinate plane, table of values, set of coordinate pairs, and mapping diagram
The range of a relation is the set of all outputs of the relation. Consider the relation R one more time. R = {(0,- 2), (1,0), (2,2) } By looking at the second coordinate of the ordered pairs, the range can be determined.

Relation Outputs Range
{(0, - 2), (1, 0), (2, 2) } - 2, 0, 2 { - 2, 0, 2 }

Depending on how a relation is represented, its range can be determined using different methods.

Graph in the coordinate plane, table of values, set of coordinate pairs, and mapping diagram

If two different inputs have the same output, it is not necessary to repeat such output when writing the range.
Example

Vincenzo's Contribution to the Games

Tadeo enjoyed displaying the relations between the ages and heights of Vincenzo's teammates so much that he wanted to go a step further and continue analyzing relations. Tadeo kept track of the scores in the first six games and Vincenzo's contribution to his team's score.

Relation R and F
In Relation R, the x-values represent the score of Vincenzo's team, while the y-values indicate the opposing team's score. Relation F shows Vincenzo's individual contribution to the score in each game. For instance, (1,11) means that Vincenzo scores 11 points in the first match.

a

What is the domain of Relation R?

What is the range of Relation R?

b

What is the domain of Relation F?

What is the range of Relation F?

Hint
a

The domain of a relation represented by a table is formed by the x-values, while its range is formed by the y-values. When writing sets, do not repeat elements.

b

To find the domain, look at the set of inputs. Similarly, to find the range, look at the set of outputs.

Solution
a

Given a relation represented by a table, the domain includes all the x-values, and the range includes all the y-values.

Relation R

By listing the numbers from least to greatest, and including each element only once, the domain and range of R are as follows. Domain ofR &= {67,77,86,93,95} Range ofR &= {63,81,82,88,89,95}

b

As done in Part A, to find the domain of F, look at the numbers inside the inputs set. To find the range, look at the outputs sets.

Relation F

By listing the numbers from least to greatest, and including each element only once, the domain and range of F are as follows. Domain ofF &= {1,2,3,4,5,6} Range ofF &= {11,17,20,27,29}

Example

Number of Spectators in Matches

Tadeo is on a roll with his basketball analyses. Other than the game itself, he has become fascinated with the crowd. Crowd-basketball.png He writes a relation showing the number of spectators in each game. In addition to that, he then writes another relation that shows the revenue from the tickets sold to these rambunctious fans.

Relation A and B
The x-coordinates of the ordered pairs in Relation P represent the number of the match, while the y-coordinates represent the number of spectators in the match. Relation Q shows the number of spectators on the x-axis and revenue in dollars on the y-axis.

a

What is the domain of Relation P?

What is the range of Relation P?

b

What is the domain of Relation Q?

What is the range of Relation Q?

Hint
a

The domain of a relation represented by a set of ordered pairs is formed by the first coordinate of each pair, while its range is formed by the second coordinates. When writing sets, do not repeat elements.

b

To find the domain, look at the set of inputs. Similarly, to find the range, look at the set of outputs.

Solution
a

Given a set of coordinate pairs, the domain includes all the first coordinates, and the range includes all the second coordinates.

R = {(18,1.75),(23,1.85),(12,1.36),(43,1.82),(47,1.82)}

By listing the numbers from least to greatest, and including each element only once, the domain and range of P are as follows. Domain ofP &= {1,2,3,4,5,6} Range ofP &= {100,110,120,125,135,145}

b

To find the domain of Q, look at the x-coordinates of the points. To find the range, look at the y-coordinates of the points.

Relation F

By listing the numbers from least to greatest, and including each element only once, the domain and range of Q are as follows. Domain ofQ &= {100,110,120,125,135,145} Range ofQ &= {500,550,600,625,675,725}

Pop Quiz

Determining the Domain and Range of Relations

The following applet displays a relation either as a table, a set of ordered pairs, a mapping diagram, or a group of coordinate points in the coordinate plane. For any given relation, determine its domain and range.

Pop Quiz

Analyzing Relations

The following applet displays two relations. Determine if they represent the same relation.

Random generator creates two relations
Closure

Representing Relations

Consider all of the diagrams given at the beginning of the lesson. Compare the information that each diagram provides and see if they correspond to each other.

Diagram I: Table

The first diagram shows a relation represented as a table. It consists of rows and columns, where each row represents an ordered pair. The first column represents the input values (x-values), and the second column represents the output values (y-values).

Diagram I: Table

To compare with other representations, ensure that the values in the table match the input-output pairs provided in other diagrams.

Diagram II: Mapping Diagram

The second diagram visually represents a relation by using arrows to connect the input values with their corresponding output values. The input values are shown on the left side, and the output values on the right side.

A diagram displaying the input values on the left with their corresponding output values on the right. Each input value has a unique output.

Notice that the arrows in the mapping diagram correctly link the input values to their respective output values. This is the same as the pairs shown in the table.

Diagram III: Set of Ordered Pairs

The third diagram is a set of ordered pairs (x, y), where x is the input value and y is the output value. Compare the pairs in the set with the corresponding values in other representations. { c(1.70,2),(1.75,5),(1.85,4), (1.91,8),(1.96,8) } This is also the same as the ordered pairs shown in the previous diagrams.

Diagram IV: Graph

Finally, the fourth diagram shows a relation as a group of coordinate points in the coordinate plane. Recall that each point consists of an x-coordinate and a y-coordinate.

Diagram IV: Graph
The points on the coordinate plane match with the input-output pairs provided in other representations. As a result the information matches across all representations. It can be concluded that they represent the same relation.



Level 1
Level 2
Level 3
Relations
Exercise 3.1
LaShay graphs the points in the table on a coordinate plane and assigns them the labels A, B, and C.

x y
3 5
3 - 3
- 5 - 3

Afterwards, she places a new point D to the graph in such a way that points A, B, C, and D together create a square on the coordinate plane.

 What are the coordinates of point D?

What is the domain of the relation formed by points A, B, C, and D?

We are given the following table of ordered pairs.

x y
3 5
3 - 3
- 5 - 3

We are asked to graph the ordered pairs. Let's do it! First, let's graph the pair with coordinates x= 3 and y= 5.

Now we will graph the other pairs from the table in a similar way.

Let's label the points A, B, and C.

Next, we want to find the coordinates of point D so that the points A, B, C, and D form a square. Notice that right now, we can connect the points A, B, and C so that they from two sides of a square.

Therefore, point D needs to be the end point of the two other sides of the square. This is possible only when the coordinates of point D are x=- 5 and y=5.

As we can see, if the points A, B, C, and D form a square, then the coordinates of point D are (- 5,5).


In the previous part, we found the coordinates of point D. Let's make a table to represent the relation formed by the four points.

x y
3 5
3 - 3
- 5 - 3
- 5 5

The domain of this relation is the set of all unique x-values. Therefore, its domain contains two elements, which are 3 and - 5. Domain: { - 5, 3 }

E
Hint & Answer
S
Solution
Two relations are shown as follows.

Relations P and Q
 Which mapping diagram represents Relation P?

Which mapping diagram represents Relation Q?

Which relation assigns each input to exactly one output?

We are given a relation, which is a set of ordered pairs of some inputs and outputs. Relation P { ( 2, 1),( 6, 5),( 2, 8),( 6, 11) } We will try to make an mapping diagram that represents this relation. Recall that the first coordinates are the inputs of the relation and the second coordinates are the outputs of the relation. Let's write the unique inputs and outputs inside two sets.

Next, we can write arrows between the inputs and outputs that correspond to each other.

The mapping diagram is done! The answer is Option B.


In this part, we will make an arrow diagram representing Relation Q. We start with identifying the coordinates of the points.

The x-coordinates are the input values and the y-values are the output values of the relation.

Relation Q
Input Output
1 2
5 6
8 2
11 6

As before, let's write all unique inputs and outputs inside two sets.

Next, we can write arrows between the inputs and outputs that are related to each other.

We created an arrow diagram that represents Relation Q! This corresponds to Option C.


Finally, we will determine in which relation each input is assigned to exactly one output. Let's take a look at the mapping diagram of Relation P that we made in Part A.

In this relation, inputs 2 and 6 have multiple corresponding outputs. Each input is assigned to two different outputs. Relation P does not have the property we want. Now, let's look at Relation Q.

In this relation, outputs 2 and 6 have multiple corresponding inputs, but that is not a problem. An output can have multiple corresponding inputs. What matters is that each input of Relation Q is assigned to exactly one output. Therefore, Relation Q has the property described in the question. The answer is then Relation Q.

A relation is called a function if it has the property we mentioned above.

When a Relation Is Also a Function? |- A relation is a function when each input is assigned to exactly one output.

In this case, Relation Q is a function because each input is assigned to exactly one output.

Relation Is it a Function?
Relation P No
Relation Q Yes


E
Hint & Answer
S
Solution

Relations

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