Relations and Functions

A mathematical concept of relation

“A relation is a set of ordered pairs.” In other words, a relation is pairing of input values with output values.

The set of the first components of the ordered pairs or the input values is called the DOMAIN. The set of the second components of the ordered pairs or the output values is called the RANGE.

EXAMPLE: R= {(6, 8), (3, 7), (1, 2), (0, 1)}

In the above example,(6,8),(3,7),(1,2),(0,1)are the ordered pairs;R is a set of these ordered pairs ,therefore R becomes relation. In the relation R,{6,3,1,0}is the domain and {8,7,2,1} is the range.

REPRESENTATION OF RELATION:

A) Using arrow diagrams or mapping:

In this form, arrows are drawn from one set (that constitutes the domain) to another set (that constitutes the range) to indicate pairings, which satisfy the given relation.

EXAMPLE: R={(1,2),(3,4),(4,5)}

In the above example, domain or set A= {1, 3, 4} and range or set B={2,4,5}

Using arrow diagrams, the relation of set A TO set B can be represented in the following manner;

A B

1 2

3 4

4 5

(B)By Graphing

To graph a relation, plot the domain on the x-axis of the graph and the range on the y-axis.

EXAMPLE:R={(1,3),(1,5),(2,3),(25),(3,5),(4,5)}

DOMAIN OF R = {1, 2, 3, 4}

RANGE OF R= {3, 5}

NOW plot the ordered pairs with the first components on the x-axis and the second components on y-axis.

C) Using x-y table

A relation can be expressed in two variables, x and y.

In an x-y table, the components of the domain stand as the variables for “x” whereas the components of the range are the variables for the “y”.

EXAMPLE: R= {(1, 3), (2, 4), (3, 6), (4, 8)}

Domain= {1, 2, 3, 4} which becomes the variables for x

Range= {3, 4, 6, 8} which becomes the variables for y

x	1	2	3	4
y	3	4	6	8

D) Using Equations

A relation can be expressed in two variables using a linear equation when all the ordered pairs in the relation exhibit a particular characteristic.

Example: R= {(1, 3), (2, 4), (3, 5), (4, 6)}

In the above relation ;{ 1, 2, 3, 4} or domain forms the values for the variable “x” &

{3, 4, 5, 6} or range forms the values for variable “y”

In each ordered pair, second component=first component +2

In other words, for every value of ‘x’, there is a ‘y’ value which is equal to 2 added to the ‘x’ value. Hence the relation can be written as:

y=x+2

FUNCTION

A relation from a set A TO set B is called a FUNCTION; if for each element of set A there is a unique (one and only one)image in set B.In other words ,a function is a relation where the first components of all the ordered pairs are different.

REMEMBER: In a function;

· Every element in set A should have its image in set B.

· An element of set A must have only one image in set B.

· The first components of the ordered pairs cannot repeat but the second components may repeat.

· Every relation which can be expressed by a linear equation in two variables represents a function.

EXAMPLES:

1) R={(1,2),3,4),(5,7),(8,10)}

The above relation is a function because the first components of all ordered pairs are different.

2) R={(1,2),(1,6),(2,3)}

The above relation is not a function because the first components are repeated.

3) A B