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 .

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?

Let R = { ( m , n ) : m , n ∈ Z : m − n is divisible by 13 ) } Let m be an arbitrary element of Z . Then , m ∈ R ⇒ m − m = 0 = 0 × 13 ⇒ m − m is divisible by 13 ⇒ ( m , m ) ∈ R So, R is reflexive on Z . Let ( m , n ) ∈ R . Then , m − n is divisible by 13 ⇒ m − n = 13 p [ Here , p ∈ Z ] ⇒ n − m = 13 ( − p ) [ Here , − p ∈ Z ] ⇒ n − m is divisible by 13 ⇒ ( n , m ) ∈ R for all m , n ∈ Z So , R is symmetric on Z . Let ( m , n ) and ( n , o ) ∈ R ⇒ m − n and n − o are divisible by 13 ⇒ m − n = 13 p and n − o = 13 q for some p , q ∈ Z Adding the above two , we get m − n + n − o = 13 p + 13 q ⇒ m − o = 13 ( p + q ) Here , p + q ∈ Z ⇒ m − o is divisible by 13 ⇒ ( m , o ) ∈ R for all m , o ∈ Z 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 } .

Let Z be the set of all integers and Z 0 be the set of all non-zero integers. Let a relation R on Z × Z 0 be defined as ( a , b ) R ( c , d ) ⇔ a d = b c , for all ( a , b ) , ( c , d ) ∈ Z × Z 0 , Prove that R is an equivalence relation on Z × Z 0 .

Let ( a , b ) be part of Z × Z 0 . Then , ( a , b ) ∈ Z × Z 0 ⇒ a b = b a ⇒ ( a , b ) R a , b So , R is reflexive on Z × Z 0 . Let ( a , b ) , ( c , d ) ∈ Z × Z 0 Then , ( a , b ) R ( c , d ) ⇒ a d = b c ⇒ c b = d a ⇒ ( c , d ) R ( a , b ) Thus , ( a , b ) R ( c , d ) ⇒ ( c , d ) R ( a , b ) So, R is symmetrical Z × Z 0 Let ( a , b ) , ( c , d ) , ( e , f ) ∈ Z × Z 0 . Then, ( a , b ) R ( c , d ) ⇒ a d = b c ( c , d ) R ( e , f ) ⇒ c f = d e Multiplying both, we get ( a d ) ( c f ) = ( b c ) ( d e ) ⇒ a f = b e ⇒ ( a , b ) R ( e , f ) So, R is transitive on Z × Z 0 So, R is an equivalence relation on Z × Z 0 .

E M B I B E

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

R D Sharma Solutions for Chapter: Relation, Exercise 1: EXERCISE

Author:R D Sharma

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Relation, Exercise 1: EXERCISE

Attempt the free practice questions on Chapter 1: Relation, Exercise 1: EXERCISE with hints and solutions to strengthen your understanding. MATHEMATICS CLASS XII VOLUME-1 solutions are prepared by Experienced Embibe Experts.

Questions from R D Sharma Solutions for Chapter: Relation, Exercise 1: EXERCISE with Hints & Solutions

MEDIUM

12th CBSE

IMPORTANT

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

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 $Z$ .

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 $\Rightarrow b - a = 5 (- p)$ 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 $(b, c) \in R$ 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 $P_{2}$ have same number of sides $}$ , is an equivalence relation.

What is the set of all elements in $A$ related to the right angle triangle $T$ with sides $3, 4$ and $5$ ?

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 1 - Relation>EXERCISE>Q 14

Let $Z$ be the set of all integers and $Z_{0}$ be the set of all non-zero integers. Let a relation $R$ on $Z \times Z_{0}$ be defined as $(a, b) R (c, d) \Leftrightarrow a d = b c$ , for all $(a, b), (c, d) \in Z \times Z_{0}$ , Prove that $R$ is an equivalence relation on $Z \times Z_{0}$ .