Question

Let $n$ be a fixed positive integer. Let a relation $R$ be defined in $I$ (the set of all integers) as follows : $aRb$ iff $n|(a-b)$, that is, iff $a$ - $b$ is divisible by $n$. Then, the relation $R$ is

• A. Reflexive only
• B. Symmetric only
• C. Transitive only
• D. An equivalence relation

Answer 1

Answer: (D)

Reflexive : Since for any integer $a$, we have $a-a=0$ is divisible by $n$. Hence, $aRa\forall a\in I$.
$\therefore R$ is Reflexive.
Symmetric : Let $aRb$. Then, by definition of $R$,
$a-b=nk$, where $k\in I$.
$b-a=(-k)n$ where $-k\in I$ and so $bRa$.
$\therefore R$ is symmetric.
Transitive : Let $aRb$ and $bRc$. Then, by definition of $R$, we have, $a-b=k_{1}n$ and $b-c=k_{2}n$, where $k_1$, $k_2\in I$.
Then it follows that
$a-c=(a-b)+(b-c)=k_{1}n+k_{2}n-(k_1+k_2)n$, where $k_1+k_2\in I$ and so $aRc$.
$\therefore R$ is transitive.
Hence, $R$ is an equivalence relation.