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Math Lesson 16.1.4 - What is a Relation in Math?
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Welcome to our Math lesson on What is a Relation in Math?, this is the fourth lesson of our suite of math lessons covering the topic of Relation and Function, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
What is a Relation in Math?
The ordered pairs in the examples of the previous lesson were chosen randomly in the sense that they were not connected through any specific rules. If these ordered pairs are connected with each other through specific rules, this is called a relation.
We represent schematically a relation in math as follows:
For example, if the set of input values is X = {-2, 0, 2, 4, 6} and the processing machine (relation) is y = 3x - 5, the set of values of the output set Y is
= 3 ∙ (-2) - 5
= -6 - 5
= -11
y2 = 3x2 - 5
= 3 ∙ 0 - 5
= 0 - 5
= -5
y3 = 3x3 - 5
= 3 ∙ 2 - 5
= 6 - 5
= 1
y4 = 3x4 - 5
= 3 ∙ 4 - 5
= 12 - 5
= 7
y5 = 3x5 - 5
= 3 ∙ 6 - 5
= 18-5
= 13
Therefore, the output set Y has the values: Y = {-11. -5, 1, 7, 13}.
Example 3
The input set A of a relation contains the elements -3, -2, -1, 0, 1, 2 and 3. The relation between the input and output sets A and B is
where x is any element taken from the input set A while y is the corresponding value from the output set B. Calculate the elements of the output set B.
Solution 3
We must insert all elements of the input set A in the relation's formula above to find the corresponding values of the output set B. Thus, we have
= √12 - (-3)2
= √12 - 9
= √3
y2 = √12 - x22
= √12 - (-2)2
= √12 - 4
= √8
y3 = √12 - x23
= √12 - (-1)2
= √12 - 1
= √11
y4 = √12 - x24
= √12 - 02
= √12 - 0
= √12
y5 = √12 - x25
= √12 - 12
= √12 - 1
= √11
y6 = √12 - x22
= √12 - 22
= √12 - 4
= √8
and
= √12 - 32
= √12 - 9
= √3
Thus, the output set B contains only four elements. They are B = {√3, √8, √11, √12}.
As you see, in the above there are some output values, which have more than one input value in correspondence. In relations, the reverse can also occur. Let's take an example to clarify this point.
Example 4
The relation between the two quantities represented by the letters x and y in a relation is given by the formula
Calculate the y-values for x = 0, 1, 2 and 3.
Solution
For x1 = 0, we have
(-3)2 + y21 = 25
9 + y21 = 25
y21 = 16
In this way, we obtain two values for y1. They are -4 and 4, as both give 16 when raised in square.
The same procedure is followed for the rest of the points as well. Thus, for x2 = 1, we have
(-2)2 + y21 = 25
4 + y21 = 25
y21 = 21
In this way, we obtain two values for y2. They are -√21 and √21, as both give 21 when raised in square.
For x3 = 2, we have
(-1)2 + y21 = 25
1 + y21 = 25
y21 = 24
In this way, we obtain two values for y3. They are -√24 and √24, as both give 24 when raised in square.
For x4 = 3, we have
02 + y21 = 25
0 + y21 = 25
y21 = 25
In this way, we obtain two values for y3. They are -5 and 5, as both give 25 when raised in square.
Therefore, the set of the output values y is y = -5, -√24, -√21, -4, 4, √21, √24, 5.
You have reached the end of Math lesson 16.1.4 What is a Relation in Math?. There are 9 lessons in this physics tutorial covering Relation and Function, you can access all the lessons from this tutorial below.
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