Home
Class 12
MATHS
Let Z be the set of all integers. A rela...

Let Z be the set of all integers. A relation R is defined on Z by xRy to mean x-y is divisible by 5. Show that R is an equivalence relation on Z.

Text Solution

AI Generated Solution

To show that the relation \( R \) defined on the set of integers \( Z \) by \( xRy \) if and only if \( x - y \) is divisible by 5 is an equivalence relation, we need to verify that it satisfies three properties: reflexivity, symmetry, and transitivity. ### Step 1: Reflexivity A relation \( R \) is reflexive if for every \( x \in Z \), \( xRx \) holds true. - For any integer \( x \), we compute \( x - x \). - We have \( x - x = 0 \). - Since 0 is divisible by 5 (as \( 0 = 5 \times 0 \)), we conclude that \( xRx \) is true for all \( x \in Z \). ...
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    AAKASH INSTITUTE ENGLISH|Exercise Try Yourself|70 Videos

  • RELATIONS AND FUNCTIONS

    AAKASH INSTITUTE ENGLISH|Exercise Assignment (Section - A) Objective Type Questions (one option is correct)|102 Videos

  • PROBABILITY

    AAKASH INSTITUTE ENGLISH|Exercise ASSIGNMENT SECTION-J (aakash challengers questions)|11 Videos

  • SEQUENCES AND SERIES

    AAKASH INSTITUTE ENGLISH|Exercise Assignment (SECTION - J) Aakash Challengers|11 Videos

Similar Questions

Explore conceptually related problems

Let n be a positive integer. Prove that the relation R on the set Z of all integers numbers defined by (x , y) in R iff x-y is divisible by n , is an equivalence relation on Z.

Let n be a positive integer. Prove that the relation R on the set Z of all integers numbers defined by (x , y) in R iff x-y is divisible by n , is an equivalence relation on Z.

Let n be a positive integer. Prove that the relation R on the set Z of all integers numbers defined by (x , y) in R iff x-y is divisible by n , is an equivalence relation on Z.

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

Show that the relation R defined by R={(a , b):a-b is divisible by 3; a , bZ} is an equivalence relation.

Show that the relation R defined by R={(a , b):a-b is divisible by 3; a , bZ} is an equivalence relation.

Show that the relation R defined by R={(a , b):a-b is divisible by 3; a , bZ} is an equivalence relation.

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

Let Z be the set of all integers and R be the relation on Z defined by R= { (a,b): a, b in Z and (a-b) is divisible by 5} . Prove that R is an equivalence relation

Let m be a given fixed positive integer. Let R={(a.b) : a,b in Z and (a-b) is divisible by m} . Show that R is an equivalence relation on Z .