Join
Topic

Relations

Estimated reading: 9 minutes 583 views Contributors

Summary: Normally relation deals with matching of elements from the first set called DOMAIN with the element of the second set called RANGE.

Relations

A relation “R” is the rule that connects or links the elements of one set with the elements of the other set.

Some examples of relations are listed below:

  1. “Is a brother of “
  2. “Is a sister of “
  3. “Is a husband of “
  4. “Is equal to “
  5. “Is greater than “
  6. “Is less than “

Normally relations between two sets are indicated by an arrow coming from one element of the first set going to the element of the other set.

Relations Between Two Sets

Find relations between two sets

The relation can be denoted as:

R = {(a, b): a is an element of the first set, b is an element of the second set}

Consider the following table

1 1441271737538

This is the relation which can be written as a set of ordered pairs {(-3, -6), (0.5, 1), (1, 2), (2, 4), (5, 10), (6, 12)}. The table shows that the relation satisfies the equation y=2x. The relation R defining the set of all ordered pairs (x, y) such that y = 2x can be written symbolically as:

R = {(x, y): y = 2X}.

Relations Between Members in a Set

Find relations between members in a set

Which of the following ordered pairs belong to the relation {(x, y): y>x}?

(1, 2), (2, 1), (-3, 4), (-3, -5), (2, 2), (-8, 0), (-8, -3).

Solution.

(1, 2), (-3, 4), (-8, 0), (8,-3).

Relations Pictorially

Demonstrate relations pictorially

For example the relation ” is greater than ” involving numbers 1,2,3,4,5 and 6 where 1,3 and 5 belong to set A and 2,4 and 6 belong to set B can be indicate as follows:-

3 1441272458358

This kind of relation representation is referred to as pictorial representation.

Relations can also be defined in terms of ordered pairs (a,b) for which a is related to b and a is an element of set A while b is an element of set B.

4 1441273481330

For example the relation ” is a factor of ” for numbers 2,3,5,6,7 and 10 where 2,3,5 and 6 belong to set A and 6,7 and 10 belong to set B can be illustrated as follows:-

5 1441273616450
6 1441273782881

Example 1

1. Draw an arrow diagram to illustrate the relation which connects each element of set A with its square.

7 1441274058229

Solution

8 1441274202039

Example 2

Using the information given in example 1, write down the relation in set notation of ordered pairs. List the elements of ordered pairs.

9 1441275673193

Example 3

As we,

10 1441275814254

Solution;

11 1441275906720

Example 4

Let X= {2, 3, 4 } and Y= {3 ,4, 5}

Draw an arrow diagram to illustrate the relation ” is less than”

12 1441276089579

Exercise 1

Let P= {Tanzania, China, Burundi, Nigeria}

Draw a pictorial diagram between P and itself to show the relation

“Has a larger population than”

2. Let A = 9,10,14,12 and B = 2,5,7,9 Draw an arrow diagram between A and B to illustrate the relation ” is a multiple of”

3.Let A = mass, Length, time and

B = {Centimeters, Seconds, Hours, Kilograms, Tones}

Use the set notation of ordered pairs to illustrate the relation “Can be measured in”

4. A group people contain the following; Paul Koko, Alice Juma, Paul Hassan and Musa Koko. Let F be the set of all first names, and S the set of all second names.

Draw an arrow diagram to show the connection between F and S

5. Let R={ (x, y): y=x+2}

Where x∈A and A ={ -1,0,1,2}

and y∈B, List all members of set B

Exercise 2

1. Let the relation be defined

13 1441280276312

Consider the following pictorial diagram representing a relation R.

14 1441280502948

Let the relation R be defined as

14 1441280502948

A relation R on sets a and B where A = 1,2,3,4,5 and B = 7,8,9,10,11,12 is defined as ” is a factor of “

16 1441282624413

Graph of a Relation

A Graph of a Relation Represented by a Linear Inequality

Draw a graph of a relation represented by a linear inequality

Given a relation between two sets of numbers, a graph of the relation is obtained by plotting all the ordered pairs of numbers which occur in the relation

Consider the following relation

22 1441285525662

The graph of R is shown the following diagram( x-y plane).

23 1441285848289

Example 5

Solved:

26 1441358901110

Note that some relations have graphs representing special figures like straight lines or curves.

Example 6

Draw the graph for the relation R= {(x, y): y = 2x +1} Where both x and y are real numbers.

Solution

The equation y = 2x +1 represents a straight line, this line passes throng uncountable points. To draw its graph we must have at least two points through which the line passes.

28 1441359619445

Graph;

29 1441359800371

Example 7

Let A = {-2,-1,0, 1, 2 } and B ={0,1,2,3,4}

Let the relation R be y= x2, where x ∈A and y∈B. Draw the graph of R

Solution

30 1441360253126

NB: When the relation is given by an equation such as y = f (x), the domain is the set containing x- values satisfying the equation and the range is the set of y-values satisfying the given equation.

Exercise 3

Test Yourself:

31 1441361431640

Quiz.

32 1441361559610
33 1441361669765

Domain and Range of a Relation

The Domain of Relation

State the domain of relation

Domain: The domain of a function is the set of all possible input values (often the “x” variable), which produce a valid output from a particular function. It is the set of all real numbers for which a function is mathematically defined.

The Range of a Relation

State the range of a relation

Range: The range is the set of all possible output values (usually the variable y, or sometimes expressed as f(x)), which result from using a particular function.

If R is the relation on two sets A and B such that set A is an independent set while B is the dependent set, then set A is the Domain while B is the Co-domain or Range.

Note that each member of set A must be mapped to at least one element of set B and each member of set B must be an image of at least one element in set A.

Consider the following relation

17 1441284260923

Example 8

Let P = 1,3,4,10 and Q = 0,4,8

Find the domain and range of the relation R:” is less than”

18 1441284671106

Example 9

As we,

19 1441284824540
20 1441284870305

Exercise 4

1. Let A = { 3,5,7,9 } and B = {1,4,6,8 } , find the domain and range of the relation “is greater than on sets A and B

21 1441285004761

4. Let X ={3, 4, 5, 6} and

Y ={2, 4, 6, 8}

Draw the pictorial diagram to illustrate the relation “is less than or equal to‘ and state its domain and range

Inequalities:

The equations involving the signs < , ≤, > or ³ are called inequalities

Eg. x

x>3 x is greater than 3

x≤ 2 x is less or equal to 2

x³ 2 x is greater or equal to 2

x > y x is greater or than y etc

Inequalities can be shown on a number line as in the following

38 1441365734971

Inequalities involving two variables:

If the inequality involves two variables it is treated as an equation and its graph is drawn in such a way that a dotted line is used for > and < signs while normal lines are used for those involving ≤ and ≥.

The line drawn separates the x-y plane into two parts/regions

The region satisfying the given inequality is shaded and before shading it must be tested by choosing one point lying in any of the two regions,

Example 10

1. Draw the graph of the relation R = {(x, y): x>y}

Solution:

x>y is the line x =y but a dotted line is used.

Graph

39 1441366747072

If you draw a graph of the relation R = {(x,y ) : x < y} , the same line is draw but shading is done on the upper part of the line.

Exercise 5

1. Draw the graph of the relation R = {(x,y ): x + y > 0}

2 .Draw the graph of the relation R = {( x ,y ) : x – y ³ -2}

3. Write down the inequality for the relation given by the following graph

40 1441367199153

4. Draw a graph of the inequality for the relation x >-2 and shade the required region.

Domain and Range from the graph

Definition: Domain is the set of all x values that satisfy the given equation or inequality.

Similarly Range is the set of all y value satisfying the given equation or inequality

Example 11

1. Consider the following graph and state its domain and range.

41 1441370381318

Solution

42 1441370514062

Example 12

State the domain and range of the relation whose graph is given below.

43 1441370636588
44 1441370715763

Inverse of a Relation

The Inverse of a Relation Pictorially

Explain the Inverse of a relation pictorially

If there is a relation between two sets A and B interchanging A and B gives the inverse of the relation.

If R is the relation, then its inverse is denoted by R-1

  • If the relation is shown by an arrow diagram then reversing the direction of the arrow gives its inverse
  • If the relation is given by ordered pair ( x, y) , then inter changing the variables gives inverse of the relation, that is (y,x) is the inverse of the relation. So domain of R = Range of R -1 and range of R = domain of R-1

Example 13

1.

34 1441362191214

The inverse of this relation is “ is a multiple of “

35 1441362286887

Inverse of a Relation

Find inverse of a relation

Example 14

Find the inverse of the relation R ={ ( x, y):x+ 3 ³ y}

Solution

R-1 is obtained by inter changing the variables x and y.

36 1441362536545

Example 15

Find the inverse of the relation

R ={ ( x , y ): y = 2x }

Solution

R ={( x , y ): y = 2x }

After interchanging the variable x and y, the equation

y = 2x becomes x = 2y

or y = ½ x

so R-1 = ( x, y ) : y = ½ x

Exercise 6

1 .Let A = 3,4,5 and B ‘= 1,4,7 find the inverse of the reaction “ is less than “ which maps an element from set A on to the element in set B

2 .Find the inverse of the relation R = {( x ,y ) : y > x – 1}

3 .Find the inverse of the following relation represented in pictorial diagram

37 1441363380997

4 .State the domain and range for the relation given in question 3 above

5. State the domain and range of the inverse of the relation given in question 1 above.

A Graph of the Inverse of a Relation

Draw a graph of the inverse of a relation

Use the horizontal line test to determine if a function has an inverse function.

If ANY horizontal line intersects your original function in ONLY ONE location, your function has an inverse which is also a function.

The function y= 3x+ 2, shown at the right, HAS an inverse function because it passes the horizontal line test.

grpap1 1472718556634
Relations - PreviousTopicNext - RelationsFunctions

Related articles

Leave a Comment


Share this Doc

Relations

Or copy link

CONTENTS

Made with ❤️ by Darasa Huru

Learning BaseContact

Thrive! English Standard Four PDF-Free Download

Do you want to see your primary school student express themselves with absolute

Excel Now! Hisabati Darasa La Tatu PDF [Download Bure]

Je, unatafuta njia rahisi ya kumsaidia mwanafunzi wako kufanya vizuri na kupenda

Thrive! Civic and Moral Standard Three PDF [Free Download]

Looking for the official TIE Civic and Moral Standard Three textbook? Access thi

Thrive! English Standard Three PDF-Free Download

Looking for the official TIE English Standard Three textbook? Grab this free dig

Learn with Joy: Social Studies Standard Three PDF Free

Need the official TIE Social Studies Standard Three textbook? Get the free digit

Ignite Curiosity! Download the Official Standard 3 Science Technology Book Free

Looking for the official TIE Science and Technology Standard Three textbook? Gra

Master Mathematics Standard Three Textbook PDF [Free Download]

Need the Mathematics Standard Three textbook? This free PDF follows the Tanzania

English Standard Five

Are you looking for the English Standard Five textbook for primary schools in Ta

Standard Seven

Excel! Social Studies Standard Four PDF [Free Download]

Need the Social Studies Standard Four textbook? This free PDF follows the Tanzan

Thrive! Civic and Moral Education Standard Four PDF Free

Download Civic and Moral Education Standard Four book PDF. Covers TIE syllabus t

Standard Six

Chat Icon Close Icon

Subscribe

×
Cancel