MATHEMATICS O LEVEL(FORM THREE) NOTES – RELATIONS

RELATIONS

A relation associates an element of one set with one or more elements of another set.

If a is an element from set A which associates another element b from set B, then the elements can be written as ordered pairs (a, b).

Thus, we can define a relation as a set of ordered pairs.

Some relations are denoted by the letter R; in set notation, a relation can be written as:

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

Example of a relation

• 1. Mwajuma is a wife of Juma.
• 2. Amina is a sister of Joyce.
• 3. y = 2x + 3
• 4. Juma is tall, Anna is short. (Not a relation)

Note

If the relation R defines the set of all ordered pairs (x, y) such that y = 2x + 3, this can be written symbolically as:

R = {(x, y) : y = 2x + 3}

PICTORIAL REPRESENTATION OF RELATIONS

Relations can be represented pictorially by:

1. Arrow diagram.
2. Cartesian graph.

Arrow diagram

An arrow diagram (arrow graph) is a representation of a relation between sets by using arrows.

Example:

Show the relation “is less than or equal to” between the members of the set {1, 2, 3, 4} by using an arrow diagram.

Solution:

R = is less than or equal to

Note: The arrow indicates that one element of one set relates to one or more elements of the other set.

The element of a set which is mapped onto another set is called the Domain of a relation. The set onto which it is mapped is called the Range of a relation.

The elements of set A above are called the domains and those of set B are called the range.

Also, we use to mean “set A is mapped onto B”.

Example 1

If x 2x, we mean “x is mapped onto 2 times x”.

When x is known, we can select values of x as 1, 2, 3, 4, 5 so the relation can be written as:

Example 2

Given that where A = {-1, 0, 2, 3, 4}. Draw a pictorial representation of the relation.

Solutions

(a) R: x 3x

Table of values

Pictorial representation

Domain and Range of a relation

Consider a relation R which is a set of all ordered pairs (x, y). The domain and range of R can be defined as follows:

Domain of R = {x : (x, y) belongs to R for some y}

Range of R = {y : (x, y) belongs to R for some x}

Note: x is called the independent variable.

y is the dependent variable.

Examples

1. Given that the relation R = {(x, y) : y is a husband of x}, find the domain and range of R.

Solution

Domain of R = {all wives}

Range of R = {all husbands}

2. Find domain and range of the relation R = {(0, 2), (0, 4), (1, 2), (3, 5)}

Solution

Domain of R = {0, 1, 3}

Range of R = {2, 4, 5}

3. Find the domain and range of the relation y = 3x² + 2

Solution

Domain = {all real numbers x}

To find the range, make x the subject.

Graphs of a relation

A graph of a relation is another way of representing a relation. The graph is drawn in the Cartesian plane and can also be called a Cartesian graph.

Examples:

1. Draw the sketch of the relation R = {(x, y) : y = 2x}, state domain and range.

Solutions:

Table of values

The graph can be obtained by plotting the ordered pairs in the x-y plane.

Domain of R = {all real numbers}

Range of R = {all real numbers}

2. Draw the graphs of the relations:

Table of values for x = y

y = -x

Note

In sketching the graph of a relation of inequalities we use:

• Dotted line (——) for < and >
• Solid line (_____) for =, ≤ and ≥

We always shade the required region for the inequalities graph.

Example

Draw the graph for the relation

Solution

The graph can be sketched as a graph of y = x.

Some points belonging to the relation R = {(x, y) : y < x} are {(2,1), (4,3), (-2,-3), (-1,-4)}.

The graph is:

THE INVERSE OF THE RELATION

The inverse of the relation R^-1 can be obtained by reversing the order in all of the ordered pairs belonging to R.

i.e. If

then

The pictorial representation for can be obtained from the picture of R by reversing the direction of all the arrows.

Pictorial representation of R

Pictorial representation of R^-1

The domain in R becomes the range of R^-1 and the range of R becomes the domain of R^-1.

The inverse of the above relation can also be found by first writing x in terms of y and then interchanging the variables. Therefore, (x, y) becomes (y, x) in the inverse relation.

Example

1. Given the relation ,

(a) Find the inverse of R.

(b) Find the domain and range of .

Solution

Interchange the variables and make y the subject:

(b)

GRAPHS OF THE INVERSE OF THE RELATION

Consider the relation .

Its inverse is .

In this case, R is the relation “less than” for all real numbers.

The graph of R and are shown as shaded regions below.

Note:

The graph of for any relation can be obtained by reflecting the graph of R about the line y = x.

Thus, we can draw the graph of when R is given by first drawing R and then reflecting it about the line y = x.

Examples

1. Draw the graph of the inverse of . Find its domain and range.
2. Draw the graph of the inverse of the relation shown in the figure below. Find its domain and range.

Solutions for question 1

The domain and range of is the intersection of the domain of the two given relations.

Solution for question 2

By using the coordinates on the boundary of R we have:

Use the ordered pairs to plot the graph of .

Domain of

Range of