PRESENTED BY

A.KRISHNA MURTY
SCHOOL ASST.(MATHS)
GOVT. HIGH SCHOOL(BOYS)
INNESEPETA
RAJAHMUNDRY
 Sets are represented in
SERBUILDER form, and
ROSTER form,
and in VENN diarams.
For more convenient to
describe a set by some property
common to all elements.
A={x/x is even natural number}
A={2,4,6,…}
 NULL SET/EMPTY SET
A null set is a set with no
elements in it.and denoted
by Ø
Ø ={ }
a boy in IX class in Girls
High School.
 SINGLETON SET
If a set contained one and only
element is known as
SINGLETON SET.
P={x/xis even prime}
P={2}
 FINITE & INFINITE SET:
While considering the
elements the process of
counting comes to an end.
Such SETs are called FINITE
SETS.
If the counting of elements
doesn’t come to an end
such SETs are called
INFINITE SETS.
 EQUAL SETS
Two sets A and B are equal
if and only if
every element in A belongs
to B
and every element in B
belongs to A
 ONE-TO-ONE
CORRESPONDENCE:
OBSERVE THE FOLLOWING

RAMESH REDDY
RAVI TEJA
RAJU SINGH
RAGHU RAVANA
 ONE-TO-ONE
CORRESPONDENCE:
OBSERVE THE FOLLOWING

RAMESH REDDY
RAVI TEJA
RAJU SINGH
RAGHU RAVANA
 ONE-TO-ONE
CORRESPONDENCE:
OBSERVE THE FOLLOWING

RAMESH REDDY
RAVI TEJA
RAJU SINGH
RAGHU RAVANA
 ONE-TO-ONE
CORRESPONDENCE:
OBSERVE THE FOLLOWING

RAMESH REDDY
RAVI TEJA
RAJU SINGH
RAGHU RAVANA
 element of A
ispairedwith one and
only one
elementofBand B
ispaired with one and
only one element of A.
So these sets
arematched ONE-TO-
ONE correspondence.
 EQUALENT SETS.
The setsA and B which
have one-to-one
correspondence are
EQUALENT SETS.
Symbolically A<-->B

or A˜B
 Cardinal number of a Set:
The number of elements in
a set is called the cardinal
number of the set.
A={1,2,3,4,5}
n(A)=5
B={2,3,4,5,6,7}
n(B)=6
 B,
if and only if every
element of A is also an
element of B.
A={1,2,3,4,5,6}
B={2,4,6}
B ⊂ ⊂ A.(B is subset of
4 A) 1 4

2 ⊂ 2 5

6 3 6
 of B,
if and only if every
element of B is also an
element of A.
A={1,2,3,4,5,6}
B={2,4,6}
A ⊃ B.(A is superset of
1 4 B)

⊃
4
2 5 2
3 6 6
Every Set is a subset of it self.
A={1,2,3,4,5}
A ⊂A
 A ⊂ B and A # B
then A is called the
PROPER SUBSET of B

1
1 A is PROPER subset of
2 B
2
3
3
4
4
5
5
6
 OBSERVE T HE FOLLOWING T ABLE
NO.OF No. of
SET ELEM E SUBSET S subse
NT S ts
A={2} 1 Ø,A 2
B={4,5} 2 Ø,B{4},{5} 4
Ø,C,(1),{2},
C={1,2,
3 {3},{1,2},{2,3}. 8
3}
{3,1}
 OBSERVE THE FOLLOWING TABLE
No. of
NO.OF
PROPER proper
SET ELEME
SUBSETS subset
NTS
s

A={2} 1 Ø 1
B={4,5} 2 Ø,{4},{5} 3
Ø,(1),{2},
C={1,2,
3 {3},{1,2},{2,3}. 7
3}
{3,1}
 POWER SET
The set contains all the
subsets of a given set A is
called
POWER SET OF A.
P(A)
If a set contains n
elements
then the POWERSET
What is SET?
COLLECTION OF
WELL DEFINED
OBJECTS is called
SET.
What is RELATION?
 RELATION is a set (having
ordered pairs) such that
for all (a,b) R/a∈A and b∈B.
R={(a,x),(a,y),(a,z),(b,x),(b,y),
(b,z),(c,x),(c,x),(c,y),(c,z)}
a x

b y

c z
 What is function?
Function is also a
relation.But
conditional Relation.
 What are the
conditions to
FUNCTIONS?
1. Every element of DOMAIN must be maping.
2. No two elements of DOMAIN have same image.
 FUNCTIONS
A function from A to B is a
relation f from A into B which
satisfies the following condition.
1.For (every element of)a∈A
there is a unique (element in)
b∈B suchthat (a,b) ∈f
2. No two ordered pairs in f have
the same first element.
f={(a,x),(b,y),(c,y)}
f x
a
y
b
z
c
f={(a,x),(b,y),(c,z)}
f
x
a
y
b
z
c
f={(a,x),(b,x),(c,x)}
f x
a
y
b
c z
f={(a,x),(b,z)}
g
x
a
y
b
z
c
g={(a,x),(b,y),(b,z),(c,z)}
g x
a
y
b
z
c

NOT FUNCTION
 TYPES OF FUNCTIONS.
1.One-One function(Injection)
2.Onto function(surjection)
3.One-One Onto function
4.Inverse of a function
5.Inverse function
6.Identity function
7.Constant function
8.Equal function.
 A function f:A→B is said to be One-One function
if no two distinct elements of A have the same
image in B.
(The second coordinates(ordinates)in One-One
function ordered pairs never repeated)

a 1
b 2 f={(a,1),(b,2),(c,3),(d,4)}
c 3
d 4
 A function f:A→B is said to be One-One function
if no two distinct elements of A have the same
image in B.
(The second coordinates(ordinates)in One-One
function ordered pairs never repeated)

a 1
f={(a,1),(b,2),(c,2),(d,5)} b 2
3
c
4
d 5
 A function f:A→B is said to be One-One function
if no two distinct elements of A have the same
image in B.
(The second coordinates(ordinates)in One-One
function ordered pairs never repeated)

a 1
f={(a,1),(b,2),(c,2),(d,5)} b 2
3
NOT c
4
d 5
 A function f:A→B is said to be Onto function if
f(A)=B.
(f is Onto if every element of B is the image of
atleast one element of A)

a 1 f={(a,1),(b,2),(c,3),(d,4)}
b 2 Domain Set=A={a,b,c,d}
c 3 Codomain Set=B={1,2,3,4}
d 4 Range Set=B
B=f(A)
 A function f:A→B is said to be Into function if
f(A)=B.
(f is Into if every element of B is the image of
atleast one element of A)

f={(a,1),(b,2),(c,2),(d,5)}
a 1
Domain Set=A={a,b,c,d} b 2

Codomain Set=B={1,2,3,4,5} 3
c
4
Range Set={1,2,5}
d 5
 A function f:A→B is said to be Bijection
if it is One-One and Onto function .

f={(a,1),(b,2),(c,3),(d,4)}

a Domain=A={a,b,c,d}
1
b 2 Codomain=Range=B
c 3 f(A)=B
d 4
f is bijection
 INVERSE OF A FUNCTION
If f is a function then the set of
ordered pairs obtained by
interchanging the first and
second coordinates of each
ordered pair in f is called the
inverse of f and
is denoted by f-1 and is read as
f-inverse.
f={(a,1),(b,2),(c,3),(d,4)}
f-1 ={(1,a),(2,b),(3,c),(4,d)}

f-1
f
a 1 1 a

b 2 2 b

c 3 3 c

d 4 4 d
-1
 INVERSE FUNCTION
If f:A→B, defined by
f={(a,b)/aε A,bε B,} is a
bijection then the Inverse
function of f denoted by f-1 is
defined as
f :B→A and f ={(b,a)/(a,b)ε f}.
-1 -1
 INVERSE FUNCTION
If f:A→B, defined by
f={(a,b)/aε A,bε B,} is a
bijection then the Inverse
function of f denoted by f-1 is
defined as
f :B→A and f ={(b,a)/(a,b)ε f}.
-1 -1
 f={(a,p),(b,q),(c,r),(d,s)}
f-1 ={(p,a),(q,b),(r,c),(s,d)}
f is Bijective function
f f-1
a p p a
b q q b
c r r c
d s s d

f =f -1
 IDENTITY FUNCTION
A function f:A→A is said to be Identity function
on A if f(x)=x ∀ x∈A, and denoted by IA

f={(a,a),(b,b),(c,c),(d,d)}
f
a a f-1 ={(a,a),(b,b),(c,c),(d,d)}
b
b
c
d c
f =f -1
d
 The function f:A→B is called
constant function the rage f
consists of only one element
I.e.∀xε A,f(x)=k; where kε B
a 1 f={(a,2)(b,2),(c,2),(d,2)}
b 2 f(A)={2}=RANGE
c 3 DOMAIN={a,b,c,d}
d 4 CODOMAIN={1,2,3,4}
 EQUAL FUNCTION
Two functions f and g are said
to be equal if and only if
(1)they are defined on the same
domain A and codomainB and
(2) f(x) =g(x) ∀xε A.
COMPOSITE FUNCTIONS.
If f:A→B and g:B→C
then the composition of f and g denoted by
gof is a mapping from A→C.
denoted by gof(x)=g{f(x)}∀xε A

f g
1 5
a
2 4
b
3 3
c
4 2
d
5 1
COMPOSITE FUNCTIONS.
gof(x)={(a,2),(b,3),(c,5),(d,1)}

f g
1 5
a
2 4
b
3 3
c
4 2
d
5 1
 IF If f:A→B and g:B→C
are ONE-ONE functions then gof:A-->C is also
ONE-ONE function.

1 a 6
2 b 7
3 c 8
4 d 9
5 e 10
 IF If f:A→B and g:B→Care two functions
and gof:A-->C is ONE-ONE function then
f is NECESSARILY ONE-ONE function.

f g
1 a
6
2 b
7
3 8 c
4 9 d
5 10 e
 IF If f:A→B and g:B→C
are two ONTO functions then gof:A-->C is also
ONTO function.

f g a
1 6
b
2 7
c
3 8
d
4 9
e
5 10
f
11
g
 IF If f:A→B and g:B→Care two functions
and gof:A-->C is ONTO function then
g is NECESSARILY ONTO function.

f g a
1 6
b
2 7
c
3 8
d
4 9
e
5 10
f
11
g
 IF If f:A→B and g:B→C
are two BIJECTIVE functions then gof:A-->C is
also BIJECTIVE function.

f g a
1 6
b
2 7
c
3 8
d
4 9
e
5 10
 IF If f:A→B and g:B→Care two functions
and gof:A-->C is BIJECTIVE function then
f is ONE-ONE AND g is necessarily ONTO
function.
f g a
1 6
b
2 7
c
3 8
d
4 9
e
5 10
f
g
 IF If f:A→B and g:B→C and h:C→D
are any three functions
ho(gof)=(hog)of
f g a 1
1 6
b 2
2 7
c 3
3 8
d 4
4 9
e 5
5 10
f
11
 IF If f:A→B is a function
and IA and IB
are two IDENTITY functions defined on A and B
respectively then

f foIA=foIB
4 1
5 2 4 1
4 1
6 3 5 2
5 2
6 6 3 3
 4
f 4 1 1
5 5 2 2
6 6 3 3

4 1
5 2
6 3
 IF If f:A→B and g:B→C are
two Bijective functions then
(gof)-1 =f-1 og-1
f
f-1
a 1 a
1
b 2 b
2
c 3 c
3
d 4 d
4
e 5 e
5
 IF If f:A→B and g:B→C are
two Bijective functions then
(gof)-1 =f-1 og-1
g
g-1
1 6 6 1
2 7 7 2
3 8 8 3
4 9 9 4
5 10 10 5
 f-1 g-1
1 a
6 1
2 b
7 2
3 c
8 3
4 d
9 4
5 e
10 5
 f g
a 1 6
b 2 7
c 3 8
d 4 9
e 5 10

gof(x)={(a,6),(b,7),(c,8),(d,9),(e,10)}
(gof)-1 ={(6,a),(7,b),(8,c),(9,d),(10,e)}
IF If f:A→B and g:B→Care
two functions suchthat
gof =IA
fog=IB
then g=f-1
 REAL FUNCTIONS
(1)If A⊆ R,then the function
f:A→B is called real variable
function.
(2)If B⊆ R, then the function
f:A→B is called a real valued
function.
(3) If both A⊆ R and B⊆ R then
the function f:A→B is called a
real function.
EVEN AND ODD FUNCTIONS
If f:A→B is a real function then f
is called even function if
xε A =>-xε A and f(-x)=f(x)
and f is called an odd function if
xε A=>-xε A and f(-x)=f(x).
 INCREASING FUNCTION
A real function f:A→B is called
an increasing function if x,yε A
and x<y =>f(x) ≤ f(y) and strictly
increasing function if x,yε A and
x<y=> f(x)<f(y).
 DECREASING FUNCTION
A real function f:A→B is called
an decreasing function if
x,yε A and x<y =>f(x) ≥ f(y)
and strictly decreasing function
if x,yε A and x<y=> f(x)> f(y).
 MONOTONIC FUNCTION
A real function f:A→B is called
Monotonic(strictly) function if
it is either
INCREASING(strictly)or
DECREASING(strictly).
A strictly Motonic function is a
ONE-ONE function.
 Important Points.
Let A & B be two non empty
sets, then the number of
relations from A→B is

2 n(A).n(B)
.
1 5
2 6
7
The number of functions that
can be defined from A→B is
[n(B)] n(A)
The number of functions that
can be defined from B→A is
[n(A)] n(B)
 The numberof CONSTANT
functions that can be defined
from A→B is n(B)
If n(A) ≤ n(B) then the number
of ONE-ONE functions from
A→B is
n(B)
P n(A)
If n(A) ≤ n(B) then the number
of MANY-ONE functions from
A→B is
[n(B)]n(A) - n(B) P n(A)
If n(A)>n(B) then the number
of ONE-ONE functions from
A→B is 0.(No one-one function
can be defined)
If n(A)>n(B) and n(B)=2
then the number of ONTO
functions from A→B are
2 n(A)
-2
 PRESENTED BY
A.KRISHNA MURTY
SCHOOL ASST.(MATHS)
G.H.S.(BOYS)
attatched to S.G.S.I.A.S.E.
INNESEPETA
RAJAHMUNDRY