Question

Let `Z` be the set of all integers and `R` be the relation on `Z` defined as `R={(a , b); a , b in Z ,` and `(a-b)` is divisible by `5.}` . Prove that `R` is an equivalence relation.
A. reflexive
B. reflexive but not symmetric
C. symmetric and transitive
D. an equivalence relation

Answer 1

Correct Answer - D
For reflexive :
(a, a)=a-a=0 is divisible by 5.
For symmetric :
If (a-b) is divisible by 5, then b-a=-(a-b) is also divisible by 5.
Thus relation is symmetric.
For transitive
If (a-b) and (b-c) is divisible by 5.
Then (a-c) is also divisible by 5
Thus relation is transitive
`therefore R` is an equivalence relation.