6 Define a relation R on as a b R if and only if a and b whe

6. Define a relation R on as (a, b) R if and only if a and b, when written out, have the same number of 5s. For example, (1752, 95) R since they both have one 5 but (1752, 505) R since 1752 has one 5 but 505 has two 5s. Is R an equivalence relation? Prove that R is an equivalence relation

Solution

1. Check for reflexivity

FOr any integer a,

aRa so R is reflexive

2. Check for symmetry

LEt, aRb ie a and b have same number of 5s so bRa

So, R is symmetric

3. Let, aRb and bRc so a ,b, c ahve same number of 5s

Hence, aRc

so R is transitive

So R is equivalence relation