MATHEMATICS: FORM THREE: Topic 1 – RELATIONS

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

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}.

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).

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:-

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.

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:-

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

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

Consider the relation below:

Let X= {2, 3, 4 } and Y= {3 ,4, 5}

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

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.

Graph;

Let the relation R be y= x^2, where x ∈A and y∈B. Draw the graph of R

Solution

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.

Test Yourself:

Quiz.

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

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

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

Solution:

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

Graph

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.

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

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

Solution

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

Solution

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

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

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

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.