Question

(10 pts) A relation R is defined on the set Z by "a R b if a - b is divisible by 5" for a, b e Z. Examine if R is an equivalence relation on Z.

Answer 1

Answer:

Yes

Step-by-step explanation:

Given that a relation R is defined on the set Z by "a R b if a - b is divisible by 5

for a, b in the set of integers.

Reflexive:

We have

for any a, $a-a=0 = 5(0)$

i.e. (a,a) is related. Hence reflexive

ii) Symmetric:

If a-b is divisible by 5, then b-a is also divisible by 5.  Hence symmetric

iii) Transitive

If $a-b = 5m and\\ b-c = 5l$

adding we get

$a-c=5(m+l)$

Hence R is transitive

It follows that R is an equivalence relation on Z