Question

Let Z be the set of all integers and let R be a relation on Z defined by $a$ R $b\iff (a-b)$ is divisible by $3$. Then, R is?

• A. Reflexive and symmetric but not transitive
• B. Reflexive and transitive but not symmetric
• C. Symmetric and transitive but not reflexive
• D. An equivalence relation

Answer 1

The correct option is D An equivalence relation

Let R={(a,b):a,b∈Z and (a−b) is divisible by 3}.

Show that R is an equivalence relation on Z.

Given:

R={(a,b):a,b∈Z and (a−b) is divisible by 3}.

$R=(a,b)$

$(a-b)$ is divisible by $3$

Reflexive

$(a,a)\Rightarrow (a-a)$ is divisible by $3$

Symmetric

$(a,b)\iff (b,a)$

$(a-b)$ is divisible by $3$

$(b-a)$ is divisible by $3$

Transitive

$(a,b),(b,c)\iff (a,c)$

$(a-b)$ is divisible by $3$

$(b-c)$ is divisible by $3$

$(a-c)$ is divisible by $3$

R is an equivalent relation on Z