Prove that the relation R on Z defined by ( a , b ) ∈ R ⇔ a − b is divisible by 5 is an equivalence relation on Z .

We have, ( a , b ) ∈ R ⇔ a − b is divisible by 5 is a relation on Z Let a be an arbitrary element of R . Then , a ∈ R ⇒ a − a = 0 = 0 × 5 ⇒ a − a is divisible by 5 ⇒ ( a , a ) ∈ R for all a ∈ Z So , R is reflexive on Z . Let ( a , b ) ∈ R ⇒ a − b is divisible by 5 ⇒ a − b = 5 p for some p ∈ Z ⇒ b − a = 5 ( − p ) ( Here , − p ∈ Z ) ⇒ b − a is divisible by 5 ⇒ ( b , a ) ∈ R for all a , b ∈ Z So , R is symmetric on Z . Let ( a , b ) and ( b , c ) ∈ R ⇒ a − b is divisible by 5 ⇒ a − b = 5 p for some Z Also , b − c is divisible by 5 ⇒ b − c = 5 q for some Z Adding the above two , we get a − c = 5 ( p + q ) ⇒ a − c is divisible by 5 (h ere , p + q ∈ Z ) ⇒ ( a , c ) ∈ R for all a , c ∈ Z So , R is transitive on Z . Hence, R is an equivalence relation on Z .

Prove that the relation R on Z defined by ( a , b ) ∈ R ⇔ a − b is divisible by 5 is an equivalence relation on Z .

We have, ( a , b ) ∈ R ⇔ a − b is divisible by 5 is a relation on Z Let a be an arbitrary element of R . Then , a ∈ R ⇒ a − a = 0 = 0 × 5 ⇒ a − a is divisible by 5 ⇒ ( a , a ) ∈ R for all a ∈ Z So , R is reflexive on Z . Let ( a , b ) ∈ R ⇒ a − b is divisible by 5 ⇒ a − b = 5 p for some p ∈ Z ⇒ b − a = 5 ( − p ) ( Here , − p ∈ Z ) ⇒ b − a is divisible by 5 ⇒ ( b , a ) ∈ R for all a , b ∈ Z So , R is symmetric on Z . Let ( a , b ) and ( b , c ) ∈ R ⇒ a − b is divisible by 5 ⇒ a − b = 5 p for some Z Also , b − c is divisible by 5 ⇒ b − c = 5 q for some Z Adding the above two , we get a − c = 5 ( p + q ) ⇒ a − c is divisible by 5 (h ere , p + q ∈ Z ) ⇒ ( a , c ) ∈ R for all a , c ∈ Z So , R is transitive on Z . Hence, R is an equivalence relation on Z .

Let n be a fixed positive integer. Define a relation R on Z as follows: ( a , b ) ∈ R ⇔ a − b is divisible by n . Show that R is an equivalence relation on Z .

We have, n a fixed positive integer and relation R on Z as ( a , b ) ∈ R ⇔ a − b is divisible by n Let a ∈ R , t hen a − a = 0 = 0 × n ⇒ a − a is divisible by n ⇒ ( a , a ) ∈ R for all a ∈ Z So , R is reflexive on Z Let ( a , b ) ∈ R Here , a − b is divisible by n ⇒ a − b = n p for some p ∈ Z ⇒ b − a = n ( − p ) ⇒ b − a is divisible by n [ p ∈ Z ⇒ − p ∈ Z ] ⇒ ( b , a ) ∈ R So , R is symmetric on Z Let ( a , b ) and ( b , c ) ∈ R Here , a − b is divisible by n and b − c is divisible by n . ⇒ a − b = n p for some p ∈ Z and b − c = n q for some q ∈ Z Adding both, we get a − b + b − c = n p + n q ⇒ a − c = n ( p + q ) ⇒ ( a , c ) ∈ R for all a , c ∈ Z So , R is transitive on Z Hence, R is an equivalence relation on Z .

Let Z be the set of integers. Show that the relation R = { ( a , b ) : a , b ∈ Z and a + b is even } is an equivalence relation on Z .

We have, R = { ( a , b ) : a , b ∈ Z and a + b is even } Let a be an arbitrary element of Z . Then , a ∈ R Clearly , a + a = 2 a is even for all a ∈ Z . ⇒ ( a , a ) ∈ R for all a ∈ Z So , R is reflexive on Z . Let ( a , b ) ∈ R ⇒ a + b is even ⇒ b + a is even ⇒ ( b , a ) ∈ R for all a , b ∈ Z So , R is symmetric on Z Let ( a , b ) and ( b , c ) ∈ R ⇒ a + b and b + c are even Now , let a + b = 2 x for some x ∈ Z and b + c = 2 y for some y ∈ Z Adding the above two , we get a + 2 b + c = 2 x + 2 y ⇒ a + c = 2 ( x + y − b ) , which is even for all x , y , b ∈ Z Thus , ( a , c ) ∈ R So , R is transitive on Z Hence, R is an equivalence relation on Z .

Show that the relation R on the set A = x ∈ Z ; 0 ≤ x ≤ 12 , given by R = ( a , b ) : a = b , is an equivalence relation. Find the set of all elements related to 1 .

Given, A = { x ∈ Z : 0 ≤ x ≤ 12 } = 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 Here, R = { ( a , b ) : a = b } for any element a ∈ A , we have ( a , a ) ∈ R , since a = a ∴ R is reflexive Now let ( a , b ) ∈ R ⇒ a = b ⇒ b = a ⇒ ( b , a ) ∈ R ∴ R is symmetric Now, let ( a , b ) ∈ R and ( b , c ) ∈ R ⇒ a = b and b = c ⇒ a = c ⇒ ( a , c ) ∈ R ∴ R is transitive. Hence, R is an equivalence relation. The elements in R that are related to 1 will be those elements from set A which are equal to 1 . Hence, the set of elements related to 1 is { 1 } .

E M B I B E

MATHEMATICS CLASS XII VOLUME-1>Chapter 1 - Relation>EXERCISE>Q 3

MEDIUM

12th CBSE

IMPORTANT

Earn 100

Prove that the relation $R$ on $Z$ defined by $(a, b) \in R \Leftrightarrow a - b$ is divisible by $5$ is an equivalence relation on $Z$ .

Important Questions on Relation

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 1 - Relation>EXERCISE>Q 4

Let

n

be a fixed positive integer. Define a relation

R

on

Z

as follows:

(a, b) \in R \Leftrightarrow a - b

is divisible by

n

. Show that

R

is an equivalence relation on

Z

.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 1 - Relation>EXERCISE>Q 5

Let

Z

be the set of integers. Show that the relation

R = {(a, b) : a, b \in Z

and

a + b

is even

}

is an equivalence relation on

Z

.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 1 - Relation>EXERCISE>Q 6

m

is said to be related to

n

if

m

and

n

are integers and

m - n

is divisible by

13

. Does this define an equivalence relation?

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 1 - Relation>EXERCISE>Q 7

Let

R

be a relation on the set

A

of ordered pair of integers defined by

(x, y) R (u, v)

if

x v = y u

. Show that

R

is an equivalence relation.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 1 - Relation>EXERCISE>Q 8

Show that the relation

R

on the set

A = \{x \in Z; 0 \leq x \leq 12\}

, given by

R = \{(a, b) : a = b\}

, is an equivalence relation. Find the set of all elements related to

1

.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 1 - Relation>EXERCISE>Q 9

Let

L

be the set of all lines in

X Y

- plane and

R

be the relation in

L

defined as

R = {L_{1}, L_{2}) : L_{1}

is parallel to

L_{2}}

. Show that

R

is an equivalence relation. Find the set of all lines related to the line

y = 2 x + 4

.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 1 - Relation>EXERCISE>Q 10

Show that the relation $R$ , defined on the set $A$ of all polygons as $R = {(P_{1}, P_{2}) : P_{1}$ and $p + q \in Z$ have same number of sides $}$ , is an equivalence relation.