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Arjun Jani
Asked: 2022-11-08T14:06:27+05:30 In:

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

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

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