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: |
|---|
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| | 11 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
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.
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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.
Which of the following diagrams can Tadeo use to illustrate the relation?
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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.
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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].](/mediawiki/images/0/08/Mljsx_Concept_Representing_Relations_3.svg)
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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.
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. { 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.
Note that by hovering over each point, the corresponding information is shown.
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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.
| 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.
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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.
a
What is the domain of Relation R?
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b
What is the domain of Relation F?
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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 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.
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}
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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.
a
What is the domain of Relation P?
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b
What is the domain of Relation Q?
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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 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.
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}
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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.
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Pop Quiz
Analyzing Relations
The following applet displays two relations. Determine if they represent the same relation.
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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.
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.
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Relations
Exercise 2.1
The mapping diagram represents the cost of booking a hotel room for a different number of days.
Which of the following is true about the mapping diagram? I. & It costs$110to book the room for two days. [0.4em] II.& If a total of $165 is paid,this means afour-day & reservation. [0.4em] III. & The cost of booking a hotel room increases & $55 per day.
{"type":"multichoice","form":{"alts":["I","II","III"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,2]}
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Consider that the number of days reserved represents the input and the cost represents the output.
We see that there is an arrow matching 2 and 110. This tells us that a two-day booking costs $110. Additionally, the third arrow matches 3 and 165 with each other. This indicates that a three-day reservation costs $165. Therefore, I is correct but II is not correct. Statement I & ✓ Statement II& * Now let's find out how the input and output values change. We can see that each input increases by 1 and the output increases by 55 dollars.
This means that the cost of reserving a hotel room increases 55 dollars per day. Therefore, III is also correct. Statement I & ✓ Statement II& * Statement III & ✓
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Solution
Dylan follows a rule to create the following mapping diagram.
Find the missing input and output to complete the mapping diagram.
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We will first try to describe the relationship between the inputs and outputs in the given mapping diagram.
To do so, we can find the difference between consecutive values. We see that each input increases by 5 units and the output decreases by 2 units at the same time.
Now, we can complete the blue diagram by adding 5 units to last value before the missing input. Also in the green diagram, we can subtract 2 units from the last value before the missing output to find the missing one. Let's do it!
In this case, we have only one output for one input. Therefore, there are not more than one possible answer. Missing Input: & 15 Missing Output: & - 11
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{"type":"choice","form":{"alts":["A","B","C","D"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":3}
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Consider that the number of pairs of socks represents the input and the cost represents the output.
The first row of the table tells us that one pair of socks costs $6. Additionally, the fourth row indicates that the cost of 4 pairs of socks is $21. Therefore, II and III are correct. Now let's find out how the input and output values change. We can easily see that each input increases by 1.
Although the cost of buying two pairs of socks is $5.5 more than the cost of buying one pair of socks, this difference is not equal to the differences between other values. This means that Statement I is false. The difference between outputs is getting smaller and smaller.
In other words, it is more profitable to buy more than one pair of socks because a pair of socks becomes cheaper on average as we increase the number of pairs of socks. For example, when we buy 4 pairs of socks, one pair costs $5.25 because 214=5.25. Therefore, IV is also correct.
We want to find the cost of 5 pairs of socks. We will follow the pattern we found in the previous part. The increase in cost is 0.5 less than the previous increase. We need to add 4 to 21 to find what we need.
The cost of 5 pairs of socks is then $25.
In the given table, the first column represents the x-values, and the second column represents the y-values.
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 row, we can see the x-coordinates and their corresponding y-coordinates. We get the ordered pairs by pairing these values. (1,6),(2,11.5),(3,16.5),(4,21),(5,25) 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 matches the graph in Option D.
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Solution
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We are asked to make a table of ordered pairs in which the x-coordinate represents the number of hours and the y-coordinate represents the amount of money Izabella earns for 1, 3, 5, and 7 hours. Number of hours & → x Amount of money & → y Since x-coordinate represents number of hours, we can fill the x column with 1, 3, 5, and 7.
| x | y |
|---|---|
| 1 | |
| 3 | |
| 5 | |
| 7 |
Now we want to find the amount of money Izabella earns mowing the lawn for the corresponding number of hours x. We know she make $12 an hour mowing the lawn. Therefore, Izabella earns $12 * x for mowing the lawn for x hours.
| x | 12* x | y |
|---|---|---|
| 1 | 12* 1 | 12 |
| 3 | 12* 3 | 36 |
| 5 | 12* 5 | 60 |
| 7 | 12* 7 | 84 |
We have created the table of ordered pairs! We can see the same table in Option C.
Recall that the range of a relation is the set of all outputs of the relation. In this case, the outputs are the y-values, which we calculated in the previous part.
| x | y |
|---|---|
| 1 | 12 |
| 3 | 36 |
| 5 | 60 |
| 7 | 84 |
Let's write the set by listing the numbers from least to greatest, and including each element only once. Range = {12,36,60,84 }
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Relations
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