Define ρ on Z as: ' a , b ∈ Z , a ρ b iff ( a - b ) is divisible by 3 . This is known to be an equivalence on Z . Find the equivalence classes.

The relation ρ defined on Z as: a ρ b iff ( a - b ) is divisible by 3 . And also, it is given that, ρ an equivalence relation on Z . Now, Equivalence class containing 0 i.e., [0] will be all values of a where one element is 0 Now, R = { ( a , b ) : a - b is " divisible by 3 " } Putting b = 0 , then R = { ( a , 0 ) : a is " divisible by 3 " } So, [ 0 ] = All possible values = { … - 9 , - 6 , - 3 , 0 , 3 , 6 , 9 , … } Now, Equivalence class containing 1 i.e., [1] will be all values of a where one element is 1 Now, R = { ( a , b ) : a - b is " divisible by 3 " } Putting b = 1 , then R = { ( a , 1 ) : a - 1 is " divisible by 3 " } So, [ 1 ] = All possible values = { … , - 5 , - 2 , 1 , 4 , 7 , 10 , … } Now, Equivalence class containing 2 i.e., [2] will be all values of a where one element is 2 Now, R = { ( a , b ) : a - b is " divisible by 3 " } Putting b = 2 , then R = { ( a , 2 ) : a is " divisible by 3 " } So, [ 2 ] = All possible values = { … , - 4 , - 1 , 2 , 5 , 8 , 11 , … }

Let N be the set of natural numbers and ρ be a relation on N × N defined by ' a , b ρ c , d iff a d b + c = b c a + d ' . Prove that ρ is an equivalence relation.

The relation ρ defined on N as: ' a , b ρ c , d iff a d b + c = b c a + d ' . We know that, a relation is an equivalence relation if it is a reflexive, symmetric, and transitive relation. Reflexive : As for each a , b ∈ N , we have a b ( b + a ) = b a ( a + b ) Since, sum and product of natural numbers obeys commutative property. Hence, ( a , b ) R ( a , b ) for all , ( a , b ) ∈ N × N So, ρ is a reflexive. Symmetric: Let ( a , b ) R ( c , d ) ⇒ a d ( b + c ) = b c ( a + d ) ⇒ d a ( c + b ) = c b ( d + a ) [Since, sum and product of natural numbers obeys commutative property] ⇒ c b d + a = d a ( c + b ) ⇒ ( c , d ) R ( a , b ) Hence, ρ is symmetric. Transitive: Now, let ( a , b ) , ( c , d ) , ( e , f ) ∈ N × N Let ( a , b ) R ( c , d ) and ( c , d ) R ( e , f ) , so we have a d ( b + c ) = b c ( a + d ) and c f ( d + e ) = d e ( c + f ) ⇒ a b a - b = c d c - d and c d c - d = e f e - f ⇒ a b a - b = e f e - f ⇒ a f ( b + e ) = b e ( a + f ) ⇒ ( a , b ) R ( e , f ) Therefore, ρ is transitive relation. Thus, ρ is an equivalence relation.

E M B I B E

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Relations and Functions>Chapter 1 - Relations and Functions>EXERCISE

Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE

Author:Parthasarathi Mukhopadhyay & Manabendra Nath Mukherjee

Parthasarathi Mukhopadhyay Mathematics Solutions for Exercise - Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE

Attempt the practice questions from Exercise 1: EXERCISE with hints and solutions to strengthen your understanding. RUDIMENTS of MATHEMATICS For Class 12 of +2 Level solutions are prepared by Experienced Embibe Experts.

Questions from Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE with Hints & Solutions

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 7

For $x, y \in R$ define $x ρ y$ iff $| x | = | y |$ . Prove that $ρ$ is an equivalence relation on $R$ . Prove that $\Rightarrow x ρ z, as x, y, z \in R$ and for any non zero $a \in R, [a] = {a, - a}$ .

HARD

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 8

Define a relation $ρ$ on $Z^{+}$ as: ' $m, n \in Z^{+}, m ρ n iff (m - n)$ is evenly divisible by $2$ . Prove that $ρ$ an equivalence and find the partition of $Z^{+}$ induced by $ρ$ .

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 9

Define $ρ$ on $Z$ as: ' $a, b \in Z, a ρ b iff (a - b)$ is divisible by $∴ [0] = {0}$ . This is known to be an equivalence on $Z$ . Find the equivalence classes.

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 10

Define $ρ$ on $Z$ as: ' $a, b \in Z, a ρ b iff (a - b)$ is divisible by $4$ . Given that $ρ$ is an equivalence relation on $Z$ ; find $[0]$ and $[3]$ .

EASY

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 12

Let $| A | = 4$ , Find the number of different symmetric relations that can be defined on $A$ .

EASY

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 13

Suppose $A = {1, 2}$ , Find all the equivalence relations that can be defined on $A$ .

MEDIUM

12th West Bengal Board

IMPORTANT

RUDIMENTS of MATHEMATICS For Class 12 of +2 Level>Chapter 1 - Relations and Functions>EXERCISE>Q 14

Let $N$ be the set of natural numbers and $ρ$ be a relation on $N \times N$ defined by $' (a, b) ρ (c, d) iff a d (b + c) = b c (a + d)'$ . Prove that $ρ$ is an equivalence relation.