# Relation and Functions Revision Notes class 12th Cbse

**Relation and Functions**

In previous class , we initiated the study of
relation and functions , where we studied about domain, co domain and range
along with different types of specific real valued function and their graphs.
all these concepts are the basics of relations and functions. In this chapter,
we will continue our study with different types of relation and functions,
composition of functions , invertible functions and binary operation.

**Concept
I Relations **

**Ordered
Pair **

A pair of elements listed in a specific order separated by comma and
enclosing within the parentheses, is called an ordered pair e.g (a,b) is an
ordered pair with a as first element and b as the second element.

**Cartesian
product **

the set of all ordered pairs (a,b) such that a `\in` A and b `\in` B is
called the Cartesian product or cross product of sets A and B ; and it is
denoted by A `\times` B . Similarly , the set of all ordered pair (b,a) such
that b `\in` B and a `\in` A is called the cartesion product or cross product
of sets B and A ; and it is denoted by B `\times` A .

thus A `\times` B = { (a,b) : a `\in` A, b `\in` B } and B
`\times` A = { (b,a) : b `\in` B, a `\in` A }

e.g if A= { 1,2,3} and B= {4,5} then A `\times` B is {(1,4) (1,5) (2,4)
(2,5)(2,5)(3,4)(3,5)(3,5)}

and B `\times` A is {(4,1)(4,2)(4,3)(5,1)(5,2)(5,2)(5,3)}

**Relation **

In the mathematics, the concept of the term 'relation' has been drawn from
the meaning of relation in English language , according to which two object or
quantities are related , if there is a recognisable connection or link between
two object or quantities

**Relation
on set A and B**

let A and B be two non-empty sets , then a relation R from set A and B is a
subset of A `\times` B i.e. R `\subset` A `times` B .

The subset R is derived by describing a relationship between first and
secound elements of ordered pairs in A`\times`B .the secound element is called
image of first elements.

**Relation
on a set**

Let A be a non-empty set, then a relation from A to itself , i.e. a subset
of A `\times` A , is called a relation on set A ( or a relation in set A ) .

example let A = {1,2,3,4} , then R = {(a,b) `\in` A`\times` A : a-b=3 } is
a relation on set A .

**Domain
, range and co domain of relations **

let us consider a relation R from set A to set B (i.e R `\subset` A
`\times` B ) such that R= {(a,b) : a `\in` A and b `\in` B } then , the set of
all first elements of the ordered pairs in R is called the domain of relation
and the set of all element of ordered pair in R is called range of relation and
set B element called co-domain

**Example
1 ****If R = {(x,y) : x+2y= 8} is a relation on a set of
natural numbers (N) , then write the domain, range and co-domain of R {All India 2014}**

solution

Given , R = {(x,y): x+2y = 8 } on a set of natural numbers,=.

consider, x+y=8, which can be re written as y = `\frac {8-x}{2}`.

now substitute value of x from natural numbers, such that y `\in`N.

on putting x=2 , we get y= `\frac {8-2}{2}` =3

on putting x=4 , we get y= `\frac {8-4}{2}` =2

on putting x=6 , we get y= `\frac {8-6}{2}` =1

thus, R = {(2,3)(4,2)(6,1)} {there is no value of x , for which y `\in`
N}

domain of R = {2,4,5}, co-domain of R=N and range of R ={3,2,1}.

**Type
of Relations **

**Empty
or Void Relation**

Relation R in set A is called an empty relation , if no element of A is
related to any element of A , i.e R= `\phi \subset A \times A`

**Universal
Relation**

Relation R in a set A is called an universal relation, if each element of A
is related to every element of A i.e R= A `\times` A.

**Identity
Relation **

Relation R in a set A is called an identity relation, if each element of A
is related to itself only and it is denoted by I , i.e I = R = {(a,a): a `\in`
A}

**Reflexive
Relation **

Relation R in set A is called reflexive relation if (a,a) `\in` R, every a
`\in` A, i.e aRa , for all a `\in` A.

**Symmetric
Relation**

Relation R in a set A is called symmetric Relation, if (a,b) `\in` R
`\Rightarrow` (b,a) `\in` R for every a,b `\in` A

**Transitive
Relation **

Relation R in a Set A is called Transitive Relation, If (a,b) `\in` R and
(b,c) `in` R `\Rightarrow` (a,c) `\in` R, For all a,b,c `\in` A

** Method
to Solve Problems Bases on Type of Relation **

In these types of problems, a set and a relation defined on that set is
given to us and we have to check or show that given relation is Reflexive or
symmetric or Transitive. For this , firstly we denote the given set as A and
given relation as R. then,

**For Reflexive,**

we have to show that for all a `\in` A (a,a) `\in` R . for this, we
take arbitrary element of set A in form of a variable and then check whether
(x,x) satisfy the given condition , then R is reflexive otherwise not.

**For symmetric,**

We have to show that for a,b `\in` A if (a,b) `\in` R , then (b,a) `\in` R.
for this, we take two arbitrary elements of set A in the form of two variables
( says x and y ) such that (x,y) `\in` R

and check (y,x) satisfy the given condition of R or not . If they satisfy
the given condition, then R is symmetric otherwise not.

**For Transitive,**

We have to show that for a,b,c `\in` A , if (a,b) `\in` R and (b,c) `\in` R
then (a,c) `\in` R . for this, we take three arbitrary elements of set A in the
form of three variables

such that (x,y) `\in` R and (y,z) `\in` R and then check whether (x,z)
satisfy the given condition, then R is Transitive otherwise not.

**Example
2 ****Check Whether the rrlation R defined in the set A =
{1,2,3,4,5,6} as R= {(x,y): y is divisible by x} is Reflexive symmetric and
Transitive. {NCERT}**

**Solution** : Given R = {(x,y): y is divisible by x }

and A = {1,2,3,4,5,6}

**Reflexive**let x `\in` A be any arbitrary element. We know that, x is divisible by x.

**Symmetric**

**Transitive**

**Example 3 ****Show that the relation R is the set {1,2,3} given by R = {(1,1)(2,2)(3,3)(1,2)(2,3)} is reflexive but neither symmetric nor transitive. {NCERT}**

**sol**let given set A = {1,2,3}

**Reflexive**here, 1,2,3 `\in` A and (1,1)(2,2)(3,3) `\in` R i.e. for all a `\in` A, (a,a) `\in` R so, R is reflexive

**Symmetric**here, (1,2) `\in` R but (2,1) `\notin` R where 1,2 `\in` A. so R is not symmetric

**Transitive**here {1,2} `\in` R and (2,3) `\in` R but (1,3) `\notin` R where 1,2,3 `\in` A so R is not transitive.

**Example 4 ****Check whether the Relation R defined in the set A ={1,2,3,...,13,14} as R = {(x,y): 3x-y=0} is reflexive, symmetric and transitive. {NCERT}**

**Sol.**Given , R = {(x,y): 3x - y = 0}

**Reflexive**

**Symmetric**

thus (1,3) `\in` R `\Rightarrow` (3,1) `\in` R

**Transitive**

**Example 5 ****Let A be the set of all lines in a plane and R be a relation in A defined by R ={(L1 , L2): L1 **⟂ **L2 }. ****(i) Show that R is symmetric but neither reflexive nor transitive. (ii) Give an example of symmetric relation in our real life **** {NCERT}**

**sol**(i) Given, R = {(L1, L2): L1 ⟂ L2} and A=set of all lines in a plane

**Reflexive**Let L `\in` A be any arbitrary element .

**Symmetric**

**Transitive**

**Example 6 Give an example of a relation, which is (i) symmetric but neither reflexive nor transitive. (ii) transitive but neither reflexive nor symmetric. (iii) reflexive and symmetric but not transitive. (iv) reflexive and transitive but not symmetric. (v) symmetric and transitive but not reflexive.**** **** {NCERT}**

**Sol:**(i) let A ={1,2,3} and defined a relation R on A as R={(1,2)(2,1)} Then , R is symmetric, as (1,2) `\in` R `\implies` (2,1) `\in` R. R is not reflexive, as 1 `\in` A but (1,1) `\notin` R, R is not transitive as (1,2) `\in` R, (2,1) `\in` R but (1,1) `\notin` R.

## Equivalence Relation

- Reflexive i.e. aRa or (a,a) `\in` R , `\forall` a `\in` A.
- symmetric i.e. aRb `\Rightarrow` bRa or (a,b) `\in` R `\Rightarrow` (b,a) `\in` R , where a,b `\in` A
- transitive i.e if aRb and bRc, then aRc or (a.b) `\in` R and (b,c) `\in` R `\Rightarrow` (a,c) `\in` R , where a,b,c `\in` A

**Example 7 Let T be the set of all triangle in a plane with R is a relation in T given by R={(T1,T2): T1 is congruent to T2 and T1 ,T2 `\in` T } show that R is an equivalence relation.**** **** {NCERT}**

**Sol**Given , T = set of all triangle in a plane and R={(T1,T2): T1 is congruent to T2 and T1 ,T2 `\in` T }

**Reflexive**

**Symmetric**

**Transitive**

### Equivalence Classes

### Example 8 Let R be the equivalence relation in the set A ={0,1,2,3,4,5} given by R = {(a,b) : 2 divides (a-b)}** then, write equivalence class {0}**** {Delhi 2014C}**

**sol :**Given A = {0,1,2,3,4,5} and R={(a,b):2 divides (a-b)} clearly, [0]={b `\in` A : (0,b) `in` R}={b `\in` A : 2 divides (0-b)}= {b `\in` A : 2 divides -b}. So . b can be 0,2,4.

### Example 9 {HOTS} let A = {1,2,3,..,9} and R be the relation in A `\times` defined by (a,b)R(c,d), if a+d = b+c for (a,b)(c,d) in A `\times` A . Prove that R is an equivalence relation and also obtain the equivalence class [(2,5)] {Delhi 2014 ; NCERT Exemplar}

**sol**Given a relation R in A `\times` A, where A={1,2,3,...,9}. defined as R = {((a,b)(c,d)) : a+d = b+c} or (a,b)R(c,d), if a+d= b+c

**Reflexive**

**Symmetric**

**Transitive**

### Example 10 Show that the relation R in the set A={1,2,3,4,5} given by R ={(a,b): |a-b| is divisible by 2} is an equivalence relation. write all the equivalence classes of R. { All India 2015C}

**Sol.**We have a relation R in set A = {1,2,3,4,5} defined as R = {(a,b): |a-b| is divisible by 2}

**Reflexive**

**Symmetric**

**Transitive**

**Concept II Functions and their Types**

**Function ( Mapping ) as a rule **

**Function (Mapping ) as a set of Ordered Pairs **

**Domain, Co-domain and Range of a function**

**Relation Between Range and Co-domain **

**Geometrical Method to Check whether a given Expression is a function or not**

**Types of Functions**

**One-One function ( Injective function )**

**Many-One Function**

**Method to check whether a function is one or many-one **

**Onto(Surjective) and Into Function**

**Method to check whether the Function is Onto or Into**

**One-One and Onto function ( Bijective )**

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