Home
Class 12
MATHS
Let R be a relation on the set A of orde...

Let R be a relation on the set A of ordered pairs of positive integers defined by `(x , y) R (u , v)`if and only if `x v = y u`. Show that R is an equivalence relation.

Text Solution

AI Generated Solution

To show that the relation \( R \) defined on the set \( A \) of ordered pairs of positive integers, where \( (x, y) R (u, v) \) if and only if \( xv = yu \), is an equivalence relation, we need to prove that \( R \) is reflexive, symmetric, and transitive. ### Step 1: Prove Reflexivity A relation \( R \) is reflexive if every element is related to itself. We need to show that for any ordered pair \( (x, y) \in A \), it holds that \( (x, y) R (x, y) \). **Proof:** Let \( (x, y) \) be any ordered pair of positive integers. We check: \[ ...
Promotional Banner

Similar Questions

Explore conceptually related problems

Let R be a relation on the set A of ordered pairs of integers defined by (x,y)R(u,v) iff xv=yu .Show that R is an equivalence relation.

Let R be a relation on the set of all real numbers defined by xRy hArr|x-y|<=(1)/(2) Then R is

Let R be the relation in the set Z of all integers defined by R= {(x,y):x-y is an integer}. Then R is

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.

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.

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

Let R be a relation on the set Q of all rationals defined by R={(a,b):a,binQ" and "a-binZ}. Show that R is an equivalence relation.

Show that the relation R on the set A of all the books in a library of a college given by R={(x,y):x and y have the same number of pages }, is an equivalence relation.