A relation associates elements from one set to elements of another set. It is represented as a set of ordered pairs. A relation can be depicted using arrow diagrams or Cartesian graphs. The domain of a relation is the set of first elements in each ordered pair, and the range is the set of second elements. The inverse of a relation reverses the order of elements in each ordered pair. The graph of a relation's inverse is its reflection across the line y=x.

1. RELATIONS
A relationassociatesanelement of one set with one or more elements of
another set.
If ''a'' is an element from set A which associatesanother element ''b'' from
set B, then the elementscan be writteninan ordered pairsas(a,b)
Thus we candefine a relationas a set of ordered pairs.
Some relationsaredenoted by letter R; in set notationa relationcan be
writtenas
R = {(a, b): a is an element of the first set, b is an element of the second set}
Example of a relation
1. 1. Mwajuma isa wife of Juma.
2. 2. Amina isa sister of Joyce.
3. 3. y = 2x + 3
4. Juma is tall, Anna is short. (Not a relation)
NOTE If therelationR defines the set of all ordered pairs(x,y) such that .
y = 2x + 3 thiscan be written symbolicallyas
R = {(x, y): y=2x +3}
PICTORIAL REPRESENTATION OF RELATIONS
Relationcan be represented pictorially;
i) Arrow diagram.

2. ii) Cartesiangraph.
Arrow diagram
An arrow diagram (arrow gramor arrow graph) is a representationofa
relationbetweensets by using the arrows.
Example:
1. Show the relation“is less thanor equal to”betweenthemembersof the
set {1, 2, 3, 4}, by using arrow diagram.
Solution:
R = is less than or equal to
Note: The arrow indicatesthat oneelement of one set relates to one or
more elements off the other set.
The element of a set which mapped onto another set is called the Domain
of a relation. Theonto set is called the Range of a relation.
The elements of set A aboveare called the domainsand those of set B
are called the range.
Also we use to mean “set A is mapped onto B”

3. Example 1
If x 2x, We mean''x is mapped onto2 timesx''.
When x is known we can select values of x as
x = 1, 2, 3, 4, 5 so the relationcanbe writtenas:-
Example 2
Given that whereA = {-1, 0, 2, 3, 4}. Draw a pictorialrepresentation
of therelation.
Solutions
(a) R: x → 3x
Table of values

4. Pictorialrepresentation
Pictorialrepresentationis
Domain and Range of a relation
Consider a relationR which is a set of all ordered pairs(x, y). The Domain
and the rangeof R canbe defined as follows.
Domainof R = {x: (x, y) belongs to R for some y}
Range of R = {y: (x, y) belongs to R for some x}

5. Note: x is called the independent variable.
y is the dependent variable.
Examples
1. 1. Given that the relationR ={(x, y): y is a husband of x},find the domain
and rangeof R
Solution
Domainof R = {all wives}
Range of R = {all husbands}
2. 2. Find domainand rangeof the relation
R = {(0, 2), (0, 4), (1, 2), (3, 5)}
Solution:
Domainof R = {0, 1, 3}
Range of R = {2, 4, 5}
3. 3. Find therange and domainof relationof
y = 3x2 + 2
Solution:
Domain= {all real numbersx}
To find the range, makex the subject.

6. Graphs of a relation
Graph of a relationis another way of representing a relation. T he graph is
drawnin the Cartesianplaneand canalso be called Cartesiangraph.
Examples:
1. 1. Draw the sketch of the relation:-
R = {(x, y): y = 2x}, statedomainand range
Solutions:-
Tableof values
The graph canbe obtained byplotting the ordered pairsin the x-y plane.
Domainof R={ all real numbers}
Range of R= {all real numbers}

7. 2. Draw the graphsof the relations
Tableof values for x = y
Tableof values x+y=0
y = -x
x 0 1 2 3 -1 -
2
-
3
y 0 -
1
-
2
-
3
1 2 3

8. Note
In sketching thegraph of a relationof inequalitiesweuse
1. Dotted line (------) for < and >
Solid line ( _____ ) for =, ≤ and ≥
We alwaysshade the required regionfor the inequalitiesgraph
Example
Draw the graph for the relation

9. Solution
The graph canbe sketched as a graph of y=x
Some points belong 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 as R-1 can be obtained by reversing the order in
all of the ordered pairs belonging to R.
i.e If

10. 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 domainin R becomesthe Ranges of
Range of

11. 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. 1. Giventhe relation
(a) Find the inverse of R
(b) Find the domainand rangeof
Solution
Interchangethevariablesand make y thesubject
(b).

12. 2.2. Given the relation
Solution
The inverse of
Writex in termsof y
GRAPHS OF THE INVERSE OF THE RELATION

13. Consider therelation
Its inverse is
In thiscase R is the relationless thanfor all real numbers,
The graph of R and
Are shown as shaded region below

14. Note: The graph of for any relationcanbe obtained byreflecting the
graph of R about theline y=x
Thus we can draw the graph of when R is given by first drawing R and
then flecting it about the line y = x
Examples
1. Draw the graph of theinverse of
Find itsDomainand range
2. Draw the graph of the inverse of the relationshown in thefigure

15. below. Find its domainand range
Solutions for question 1
The domain and range of is the intersection of the domain of the two
given relations

16. Solution for question 2
By using the coordinateon theboundaryof R we have
Use the ordered pair to plot the graph of
Domainof
Range of