function sets

8/9/2019 Function Sets

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,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,}


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NULL SET/EMPTY SETA null set is a set with noel ements i n i t .and denotedby

={ }a boy in IX class in GirlsHigh School.


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SINGLETON SET

If a set contained one and only

element is known as

SINGLETON SET.P={x/xis even prime}

P={2}


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FINITE & INFINITE SET:While considering theelements the process ofcounti ng comes to an end.Such SETs are called FINITESETS.If the counting of elementsdoesnt come to an endsuch SETs are called


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EQUAL SETSTwo sets A and B are equalif and only ifevery element in A belongsto Band every element in Bbelongs to A


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CORRESPONDENCE:

OBSERVE THE FOLLOWING

RAMESH

RAVI

RAJU

RAGHU

REDDY

TEJA

SINGH

RAVANA


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CORRESPONDENCE:


RAMESH

RAVI

RAJU

RAGHU

REDDY

TEJA

SINGH

RAVANA


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CORRESPONDENCE:


RAMESH

RAVI

RAJU

RAGHU

REDDY

TEJA

SINGH

RAVANA


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CORRESPONDENCE:


RAMESH

RAVI

RAJU

RAGHU

REDDY

TEJA

SINGH

RAVANA


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element of Aispairedwith one and

only oneelementofBand B

ispaired with one andonly one element of A.

So these setsarematched ONE-TO-ONE correspondence.

EQUALENT SETS


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EQUALENT SETS.The setsA and B which

have one-to-onecorrespondence are

EQUALENT SETS.Symbolically AB

or AB


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Cardinal number of a Set:The number of elements ina set is called the cardinalnumber of the set.A={1 ,2 ,3 ,4 ,5}n(A)=5B ={2,3,4 ,5,6 ,7}

B


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B,if and only if every

element of A is also anelement of B.

A={1,2,3,4,5,6}B={2,4,6}B A.(B is subset of

A) 1 42 5

3 6

4

2

6

f B


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of B,if and only if every

element of B is also anelement of A.

A={1,2,3,4,5,6}B={2,4,6}A B.(A is superset of

B)1 42 5

3 6

4

2

6


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Every Set is a subset of it self.

A={1,2,3,4,5}A A


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A B and A # Bthen A is called thePROPER SUBSET of B1

2

3

4

5

6

1

2

3

4

5

A is P ROP E R su bset ofB


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SET

NO.OF

ELEME

NT S

SUBSETS

No. of

subse

tsA={2} 1 ,A 2

B={4,5} 2 ,B{4},{5} 4

C={1,2,

3}3

,C,(1),{2},

{3},{1,2},{2,3

{3,1}

8

OBSERVE T HE FOLLOWING T ABLE


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SET

NO.OF

ELEME

NTS

PROPER

SUBSETS

No. of

proper

subset

s

A={2} 1 1

B={4,5} 2 ,{4},{5} 3

C={1,2,

3}3

,(1),{2},

{3},{1,2},{2,3}.

{3,1}

7

OBSERVE THE FOLLOWING TABLE


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POWER SETThe set contains all thesubsets of a given set A iscalled

POWER SET OF A.P(A)If a set contai ns nelementsthen the POWERSET


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What is SET?


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COLLECTION OF

WELL DEFINEDOBJECTS is called

SET.


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What is RELATION?


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RELATION is a set (having

ordered pairs) such that

for all (a,b) R/aA and bB.R={(a,x),(a,y),(a,z),(b,x),(b,y),

(b,z),(c,x),(c,x),(c,y),(c,z)}

a

b

c

x

y

z


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What is function?

Function is also arelation.But

conditional Relation.


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What are theconditions to

FUNCTIONS?1. Every element of DOMAIN must be maping.

2.No two elements of DOMAIN have same image.

FUNCTIONS


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FUNCTIONS

A function from A to B is a

relation f from A into B whichsatisfies the following condition.

1.For (every element of)a

Athere is a unique (element in)

bB suchthat (a,b) f

2. No two ordered pairs in f have

the same first element.


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a

b

c

x

y

z

f

f={(a,x),(b,y),(c,y)}


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a

b

c

x

y

z

f

f={(a,x),(b,x),(c,x)}


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a

b

c

x

y

z

g

f={(a,x),(b,z)}


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a

b

c

x

y

z

g

NOT FUNCTION

g={(a,x),(b,y),(b,z),(c,z)}


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TYPES OF FUNCTIONS.

1.One-One function(Injection)

2.Onto function(surjection)

3.One-One Onto function

4.Inverse of a function5.Inverse function

6.Identity function

7.Constant function

8.Equal function.

A f ti f A B i id t b O O f ti


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A function f:AB is said to be One-One functionif no two distinct elements of A have the same

image in B.

(The second coordinates(ordinates)in One-Onefunction ordered pairs never repeated)

a

b

c

d

1

2

3

4

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



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image in B.


a

b

c

d

1

2

3

4

5

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



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image in B.


a

b

c

d

1

2

3

4

5

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

NOT


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A function f:AB is said to be Onto function iff(A)=B.

(f is Onto if every element of B is the image ofatleast one element of A)

a

b

c

d

1

2

3

4

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

Domain Set=A={a,b,c,d}

Codomain Set=B={1,2,3,4}

Range Set=B

B=f(A)


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A function f:AB is said to be Into function iff(A)=B.

(f is Into if every element of B is the image ofatleast one element of A)

a

b

c

d

1

2

3

4

5

Domain Set=A={a,b,c,d}

Codomain Set=B={1,2,3,4,5}

Range Set={1,2,5}

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


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A function f:AB is said to be Bijectionif it is One-One and Onto function .

a

b

c

d

1

2

3

4

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

Domain=A={a,b,c,d}

Codomain=Range=B

f(A)=B

f is bijection


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INVERSE OF A FUNCTION

If f is a function then the set of

ordered pairs obtained by

interchanging the first and

second coordinates of eachordered pair in f is called the

inverse of f andis denoted by f-1 and is read as

f-inverse.


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-1

a

b

c

d

1

2

3

4

a

b

c

d

1

2

3

4

ff-1

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

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


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INVERSE FUNCTIONIf f:AB, defined by

f={(a,b)/a A,b B,} is a

bijection then the Inverse

function of f denoted by f-1 is

defined asf-1:BA and f-1={(b,a)/(a,b) f}.


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INVERSE FUNCTIONIf f:AB, defined by

f={(a,b)/a A,b B,} is a

bijection then the Inverse

function of f denoted by f-1 is

defined asf-1:BA and f-1={(b,a)/(a,b) f}.

f {( ) (b ) ( ) (d )}


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f={(a,p),(b,q),(c,r),(d,s)}

f-1={(p,a),(q,b),(r,c),(s,d)}

f is Bijective function

ab

c

d

pq

r

s

f

ab

c

d

pq

r

s

f-1

f f-1 IDENTITY FUNCTION


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IDENTITY FUNCTION

A function f:AA is said to be Identity functionon A if f(x)=x xA, and denoted by IA

ab

c

d

f

a

b

c

d

f={(a,a),(b,b),(c,c),(d,d)}

f-1={(a,a),(b,b),(c,c),(d,d)}

f =f-1


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The function f:AB is calledconstant function the rage fconsists of only one element

I.e.x A,f(x)=k; where k Ba

b

c

d

1

2

3

4

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

f(A)={2}=RANGEDOMAIN={a,b,c,d}

CODOMAIN={1,2,3,4}

EQUAL FUNCTION


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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) x A.

COMPOSITE FUNCTIONS


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COMPOSITE FUNCTIONS.If f:AB and g:BC

then the composition of f and g denoted by

gof is a mapping from AC.denoted by gof(x)=g{f(x)}x A

a

b

c

d

1

2

3

4

5

5

4

3

2

1

fg


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COMPOSITE FUNCTIONS.

gof(x)={(a,2),(b,3),(c,5),(d,1)}

a

b

c

d

1

2

3

4

5

5

4

3

2

1

fg


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IF If f:AB and g:BCare ONE-ONE functions then gof:A-->C is also

ONE-ONE function.

1

2

3

4

5

a

b

c

d

e

6

7

8

9

10


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IF If f:AB and g:BCare two functionsand gof:A-->C is ONE-ONE function then

f is NECESSARILY ONE-ONE function.

1

2

3

4

5

a

b

c

d

e

67

8

9

10

f g

IF


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IF If f:AB and g:BCare two ONTO functions then gof:A-->C is also

ONTO function.

a

b

c

d

e

f

g

6

7

8

9

10

11

f g1

2

3

4

5

IF


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IF If f:AB and g:BCare two functionsand gof:A-->C is ONTO function then

g is NECESSARILY ONTO function.

a

b

c

d

e

f

g

6

7

8

9

10

11

f g1

2

3

4

5


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IF If f:AB and g:BCare two BIJECTIVE functions then gof:A-->C is

also BIJECTIVE function.

a

b

c

d

e

6

7

8

9

10

f g1

2

3

4

5

IF


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IF If f:AB and g:BCare two functionsand gof:A-->C is BIJECTIVE function then

f is ONE-ONE AND g is necessarily ONTOfunction.

a

b

c

d

e

f

g

6

7

8

910

f g1

2

3

4

5


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IF If f:AB and g:BC and h:CDare any three functions

ho(gof)=(hog)of

a

b

c

d

e

f

6

7

8

910

11

f g1

2

3

4

5

1

2

3

4

5


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IFIf f:AB is a functionand IA and IBare two IDENTITY functions defined on A and B

respectively then

foIA=foIB4

5

6

f

1

2

3

4

5

6

4

5

6

1

2

3

1

2

3


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4

5

6

f

1

2

3

4

5

6

4

5

6

1

2

3

1

2

3


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IF If f:AB and g:BC aretwo Bijective functions then

(gof)-1=f-1og-1

a

b

c

d

e

1

2

34

5

f

1

2

3

4

5

a

b

cd

e

f-1


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IF If f:AB and g:BC aretwo Bijective functions then

(gof)-1=f-1og-1

1

2

3

4

5

6

7

89

10

gg-1

6

7

8

9

10

1

2

3

4

5


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6

7

8

9

10

1

2

3

4

5

g-1

a

b

c

d

e

f-1

1

2

3

4

5


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a

b

c

d

e

1

2

3

4

5

6

7

8

9

10

f g

gof(x)={(a,6),(b,7),(c,8),(d,9),(e,10)}

(gof)-1={(6,a),(7,b),(8,c),(9,d),(10,e)}


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IF If f:AB and g:BCaretwo functions suchthat

gof =IA

fog=IBthen g=f-1

REAL FUNCTIONS


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REAL FUNCTIONS

(1)If AR,then the functionf:AB is called real variable

function.

(2)If BR, then the functionf:AB is called a real valued

function.

(3) If both AR and BR thenthe function f:AB is called a

real function.


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INCREASING FUNCTION

A real function f:AB is calledan increasing function if x,y A

and xf(x) f(y) and strictlyincreasing function if x,y A and

x f(x)


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DECREASING FUNCTION

A real function f:AB is calledan decreasing function if

x,y A and xf(x) f(y)and strictly decreasing function

if x,y A and x f(x)> f(y).

MONOTONIC FUNCTION


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MONOTONIC FUNCTION

A real function f:AB is calledMonotonic(strictly) function if

it is either

INCREASING(strictly)orDECREASING(strictly).

A strictly Motonic function is aONE-ONE function.

Important Points.


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p

Let A & B be two non empty

sets, then the number ofrelations from AB is

2n(A).n(B) .12

5

6

7


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The number of functions that

can be defined from AB is[n(B)]n(A)


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The number of functions that

can be defined from BA is[n(A)]n(B)


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The numberof CONSTANT

functions that can be definedfrom AB is n(B)


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If n(A) n(B) then the numberof ONE-ONE functions from

AB isn(B) P

n(A)


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If n(A) n(B) then the numberof MANY-ONE functions from

AB is[n(B)]n(A) - n(B) P n(A)


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If n(A)>n(B) then the number

of ONE-ONE functions from

AB is 0.(No one-one functioncan be defined)


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PRESENTED BYA.KRISHNA MURTY

SCHOOL ASST.(MATHS)G.H.S.(BOYS)

attatched to S.G.S.I.A.S.E.

INNESEPETA

RAJAHMUNDRY

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