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Class – XII, CBSE MB1201: Relations And Functions

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Define a relation R from a set A to set B.

Class – XII,CBSE MB1201: Relations And Functions

Important Definitions

Topic: Relations



Define a relation R from a set A to set B.



Topic: Relations

A relation R from a set A to set B is

a subset of A × B. If R is a relation

from a set A to set B and, if

, , then we say that is

related to under the relation R

and we express this fact by

wri

a b R a

b

ting as R . a b



Define a relation R in set A.



Topic: RelationsClass – XII,CBSE MB1201: Relations And Functions


Define a relation R in set A.


A relation R in a set A is a subset of .A A




What is empty relation?



Topic: Types of RelationsClass – XII,CBSE MB1201: Relations And Functions


What is empty relation?



Topic: Types of Relations

A relation R in a set A is called

, if no element of A is related

to any element of A, i.e., .

empty

relation

R A A



What is universal relation?





What is universal relation?




A relation R in a set A is called

universal relation, if each element

of A is related to every element of

A, i.e., R = A × A.

Both the empty relation and the

universal relation are called as

trivial relations.



Define different types of relations in a set.





Define different types of relations in a set.




1 2

2 1 1 2

1 2

2 3 1 3

1 2

A relation R in a set A is called:

Reflexive, if , , for every

,

Symmetric, if , implies

that , , for all , .

Transitive, if , and

, implies that , ,

for all , ,

a a R

a A

a a R

a a R a a R

a a R

a a R a a R

a a a

3 .A



What is equivalence relation?





What is equivalence relation?




A relation R in a set A is said to be an

if R is reflexive,

symmetric and transitive.

equivalence relation



Define one-one and many-one function.



Topic: Types of Functions Class – XII,CBSE MB1201: Relations And Functions





Topic: Types of Functions

1 2

1 2 1 2

A function : is defined to be

one-one (or injective), if the images

of distinct elements of X under are

distinct, i.e., for every , ,

implies .

Otherwise, is called many-one.

f X Y

f

x x X

f x f x x x

f







1 4

2 3

The function and in the above

Figure (i) and (iv) are one-one and

the function and in Figure (ii)

and (iii) are many-one.

f f

f f


Define onto function.









A function : is said to be

( ), if every element of Y

is the image of some element of X

under , i.e., for every , there exists

an element in X such that .

f X Y onto

or surjective

f y Y

x f x y






3 4

1

2

1 1

The function and in Figure (iii), (iv)

are onto and the function in Figure (i)

is not onto as elements and in are

not the image of any element in under .

f f

f

e f X

X f


Define bijective function.





Define bijective function.



Topic: Types of FunctionsClass – XII,CBSE MB1201: Relations And Functions

A function : is said to be

one-one and onto (or bijective), if

is both one-one and onto.

f X Y

f

4The function in Figure (iv) is one-one

and onto.

f


Define composition of functions.



Topic: Composition of FunctionsClass – XII,CBSE MB1201: Relations And Functions






Let : and : be two

functions. Then the composition of

and , denoted by , is defined

as the function : given by

, .

f A B g B C

f g g f

g f A C

g f x g f x x A







What is invertible function?



Topic: Invertible FunctionClass – XII,CBSE MB1201: Relations And Functions


What is invertible function?



Topic: Invertible FunctionClass – XII,CBSE MB1201: Relations And Functions

1

A function : is said to be inver-

tible, if there exists a function :

such that and . The

function is called the and is

denoted by , here d

X Y

X

f X Y

g Y X

g f I f g I

g inverse of f

f I

Invertible function :

enotes the identity

function on the set X ,and denotes the

identity function on the set Y.YI


Define different binary operations on a set.



Topic: Binary OperationsClass – XII,CBSE MB1201: Relations And Functions






A binary operation on a set A is a

function : . We denote ( , ),

the value of the function on the element

(a, b) by .

A binary operation on the set A is

called commutative, if ,

for eve

A A A a b

a b

a b b a

ry , .a b A






A binary operation on the set a is

said to be associative if,

, , , , .

Given a binary operation :

an element , if it exists, is called

identity for the operation , if

,

a b c a b c a b c A

A A A

e A

a e a e a a

.A






Given a binary operation :

with the identity element in A, an

element is said to be with

respect to the operation , if there exists an

element in A such that

and

A A A

e

a A invertible

b a b e b a

b

1

is called the inverse and is

denoted by .

of a

a


Important Facts



.R X X

Fact # 1

Empty relation is the relation R in a set X given by


Important Facts



.R X X

Fact # 2

Universal relation is the relation R in a set X given by


Important Facts



, .a a R a X

Fact # 3

Reflexive relation R in X is a relation with


Important Facts



,a b R , .b a R

Fact # 4

Symmetric relation R in X is a relation satisfying implies


Important Facts



,a b R ,b a R

, .a c R

Fact # 5

Transitive relation R in X is a relation satisfying and implies that


Important Facts


Topic: Equivalence RelationsClass – XII,CBSE MB1201: Relations And Functions

Fact # 6

Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.


Important Facts



a a X

Fact # 7

Equivalence class containing for an equivalence relation R in aset X is the subset of X containing all elements b related to a.


Important Facts



:f X Y

1 2 1 2 1 2, .f x f x x x x x X

Fact # 8

A function is one-one (or injective) if


Important Facts



:f X Y ,y Y x X

.f x y

Fact # 9

A function is onto (or surjective) if given any such that


Important Facts



:f X Y

Fact # 10

A function is one-one and onto (or bijective), if f is bothone-one and onto.


Important Facts



:f A B :g B C

:g f A C .gof x g f x x A

Fact # 11

The composition of functions and is the function given by


Important Facts


Topic: Invertible FunctionsClass – XII,CBSE MB1201: Relations And Functions

:f X Y :g Y X Xgof I

.Yfog I

Fact # 12

A function is invertible if such that and


A function is invertible if and only if f is one-one and onto. That is “f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible”. This fact significantlyhelps for proving a function f to be invertible by showing that f is one-one and onto, specially when the actual inverse of f is not to bedetermined.

Important Facts


Topic: Invertible FunctionsClass – XII,CBSE MB1201: Relations And Functions

:f X Y

Fact # 13


Important Facts



:f X X

Fact # 14

Given a finite set X, a function is one-one (respectively onto) if and only if f is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for infinite set.


Important Facts



A A

Fact # 15

A binary operation on a set A is a function from to A.


Important Facts



e X : ,X X .a e a e a X

Fact # 16

An element is the identity element for binary operationIf


An element is invertible for binary operation if there exists such that where, e is the identity forthe binary operationThe element b is called inverse of and is denoted by .

Important Facts



a X : ,X X X b X a b e b a

.1a

Fact # 17

a


Important Facts



,a b b a a b

Fact # 18

An operation on X is commutative if in X.


Important Facts



An operation on X is associative if , , in X.a b c a b c a b c

Fact # 19


Fact # 20

Important Facts



Given an arbitrary equivalence relation R in an arbitrary set X, R divides X

into mutually disjoint subsets called partitions or subdivisions of X

satisfying the following:

All elements of are re

i

i

A

A lated to each other, for all .

No element of is related to any element of , .i j

i

A A i j


Fact # 20

Important Facts


Topic: Equivalence ClassesClass – XII,CBSE MB1201: Relations And Functions

and , .

The subsets are called .

j i j

i

A X A A i j

A equivalence classes


Fact # 21

Important Facts



If R and S are two equivalence relations on a set A, then is also an

equivalence relation on A.

R S


Fact # 22

Important Facts



The union of two equivalence relations on a set is not necessarily an

equivalence relation on the set.


Fact # 23

Important Facts



1

1

If R is an equivalence relation on a set A, then defined as

, : , is also an equivalence relation on A.

R

R b a a b R


Fact # 24

Important Facts



For an arbitrary finite set X, a one-one function : is necessarily

onto and an onto map : is necessarily one-one. For an infinite

set, this may not be true. In fact, this is a characteristic d

f X X

f X X

ifference between

a finite and an infinite set.


Fact # 25

Important Facts


Topic: Identity Element for the Operations


Zero is the identity for the addition operation on R but it is not identity

for the addition operation on N, as 0 . In fact the addition operation

on N does not have any identity.

N


Fact # 26

Important Facts




For the addition operation : , given any , there exists

in R such that a + = 0 (identity for '+') .

a

a a a a

R R R R


Fact # 27

Important Facts




For the multiplication operation on R, given any 0 in R, we can choose

1 1 1in R such that 1 (identity for '×') = .

a

a aa a a


Important Formulae

If : , : and are functions, then

.

f X Y g Y Z Z S

h g f h g f




Important Formulae

1 1 1

If : and : be two invertible functions. Then is also

invertible with .

f X Y g Y Z g f

g f f g




Important Procedures



Let A be the set of all straight lines in the plane. Define a relation 'R' on A.




Topic: RelationsClass – XII,CBSE

If A is the set of all straight lines in the plane.

A relation 'R' on A can be defined as follows:

or , if and only if a is parallel to be.

We can easily verify that this relation is an equivale

aRb a b R

nce relation on A.

Thus, we can define its equivalence class denoted by as follows:

: , .

a A a

a b A a b R

MB1201: Relations And Functions





Topic: RelationsClass – XII,CBSE

We note the following

, , or .

For all , , the union of all equivalence classes of elements of

A is equal to A.

a a A

a b A a b a b

a A a




How to use operation table for the binary operation?Important Procedures


Topic: Operation Table for the Operation



How to use operation table for the binary operation?Important Procedures


Topic: Operation Table for the Operation

Class – XII,CBSE

When number of elements in a set A is small, we can express a binary

operation on the set A with the help of a table called the operation table

for the operation . For example consider A = {1, 2, 3}. T

hen, the operation

on A defined as : given by ( , ) max { , } can be expressed

by the following operation table Here, (1, 3) = 3, (2, 3) = 3, (1, 2) = 2.

A A A a b a b


3* 1 2

31 1 2

32 2 2

3 3 3 3


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