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Pre-Algebra View details

1. Relations

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eCourses /

Pre-Algebra /

Chapter 8

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.

Lesson Settings & Tools

	11 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

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?

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.

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.

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.

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:

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

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.

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.

{(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.

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.

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.

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.

By listing the numbers from least to greatest, and including each element only once, the domain and range of

R

are as follows.

Domain of R Range of R = {67, 77, 86, 93, 95} = {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.

By listing the numbers from least to greatest, and including each element only once, the domain and range of

F

are as follows.

Domain of F Range of F = {1, 2, 3, 4, 5, 6} = {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.

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.

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.

By listing the numbers from least to greatest, and including each element only once, the domain and range of

P

are as follows.

Domain of P Range of P = {1, 2, 3, 4, 5, 6} = {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.

By listing the numbers from least to greatest, and including each element only once, the domain and range of

Q

are as follows.

Domain of Q Range of Q = {100, 110, 120, 125, 135, 145} = {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.

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).

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.

{(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.

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

Exercises

Consider a relation represented by the following mapping diagram.

Which set of ordered pairs represents the relation?

The table shows the relationship between $x -$ and $y -$ values.

Input, $x$	$5$	$- 5$	$3$	$- 4$	$4$
Output, $y$	$- 5$	$5$	$- 3$	$4$	$- 4$

Which graph represents the same relation?

A mapping diagram illustrates how each element in the input is paired with an element in the output. An ordered pair is a set of numbers written in the form (x,y), where x is a number from the input and y is a number from the output.

Looking at where the arrows start and end, we can state the x-coordinates and their corresponding y-coordinates of the ordered pairs. We get the ordered pairs by pairing these values. ( 1, 1), ( 2, 4), ( 3, 9), ( 4, 16),( 5, 25) Since a relation can also be thought of as a set of ordered pairs, we write these ordered pairs between curly braces. { ( 1, 1), ( 2, 4), ( 3, 9), ( 4, 16),( 5, 25) } This set matches the set in Option A.

In the given table, the first row represents the input values (x-values), and the second row represents the output values (y-values).

Input, x	5	- 5	3	- 4	4
Output, y	- 5	5	- 3	4	- 4

An ordered pair is a set of numbers written in the form (x,y), where x is a number from the input and y is a number from the output. Looking at each column, we can state the x-coordinates and their corresponding y-coordinates. We get the ordered pairs by pairing these values. ( 5, - 5), ( - 5, 5), ( 3, - 3), ( - 4, 4),( 4, - 4) Let's plot these points on a coordinate plane. We will place the inputs along the horizontal axis and the outputs along the vertical axis.

This graph matches the graph in Option C.

Consider the set of ordered pairs that represent the number of people in a park.

{(1, 26), (2, 30), (3, 30), (4, 36), (5, 38)}

The input values indicate the day and the output values indicate the number of people in the park on that day. Which mapping diagram represents the relation?

We are given a set of ordered pairs representing the number of people in a park. { (1,26),(2,30),(3,30),(4,36),(5,38) } To express the set of given ( x, y) ordered pairs as a mapping diagram, we will list all of the unique x-values in one vertical list and all unique y-values in another vertical list. Then, we can use arrows to connect corresponding x- and y-values.

Notice that one element in the range have mapped more than one element in the domain. We did not write the same element, 30, in the range twice, instead we drew two arrows pointing to that element. Among the given options, the diagram in Option B matches the diagram we drew.

The graph represents the number of teams in each round of a football tournament.

What is the domain of the relation?

What is the range of the relation?

We are given a graph that shows the number of teams in each round of a football tournament. To determine the domain of the relation, we will identify the x-coordinates of the points. We will trace along the grid lines from each point to the x-axis.

The domain of this relation is the set of all unique x-values, which are 1, 2, 3, and 4. Domain: {1,2,3,4 }

The y-coordinates of the points will be the elements of the range of the relation. Let's identify the y-coordinates of the points.

The y-coordinates are 4, 8, 16, and 32. Therefore, the range of the relation consists of those numbers. Range: {4,8,16,32 }

The following table shows the amount of garbage that was produced in England each year between $2017$ and $2022 .$

Year, $t$	$2017$	$2018$	$2019$	$2020$	$2021$	$2022$
Garbage, $G$ (million tons)	$23.6$	$23$	$23$	$23$	$23.8$	$23.7$

The table represents a relation which assigns an input $t$ to the total amount of garbage $G$ produced in that year.

What is the domain of the relation?

What is the range of the relation?

We want to express the domain of a relation represented as a table. In the table, the input values are represented by t, a year between 2017 and 2022. This means that the domain consists of those values.

We see that the domain is the set of all unique t-values. Domain {2017,2018,2019,2020,2021,2022 }

From the table, we see that each element in the first row is paired with an element in the second row. Those values will form the range of our relation.

We see that one of the values appears three times in the second row. We include it only once when forming the set of range. Let's form the set by listing the numbers from least to greatest, and including each element only once. Range {23,23.6,23.7,23.8 }

Relations

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