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, $a - b = k_1n$ and $b - c = k_2n$, where $k_1$, $k_2 \in I$.
Then it follows that
$a - c = (a - b) + (b - c) = k_1n + k_2n - (k_1 + k_2)n$, where $k_1 + k_2 \in I$ and so $aRc$.
$\therefore R$ is transitive.
Hence, $R$ is an equivalence relation.