Question

Let $n$ be a fixed positive integer. Defined a relation $R$ on the set $Z$ of integers by, $aRb\iff n|a-b$. Then what is the relation $R$?

• A. reflexive
• B. symmetric
• C. transitive
• D. equivalence

Answer 1

The correct option is D equivalence

A relation $R$ is an equivalnce relation if $R$ is reflexive,symmetric and transitive

A relation $R$ in a set $A$ is called reflexive, if $(a,a)\in R$ for every $a\in A$

A relation $R$ in a set $A$ is called symmetric, if $(a_1,a_2)\in R\Rightarrow (a_2,a_1)\in R$ for $a_1,a_2\in A$

A relation $R$ in a set $A$ is called transitive, if $(a_1,a_2)\in R$ and $(a_2,a_3)\in R\Rightarrow (a_1,a_3)\in R$ for all $a_1,a_2,a_3\in A$

Relation $R$ on $Z,n$ a fixed integer.

$aRb$ if and only if $a-b$ is divisible by $n.$

$aRa$ is true since $a-a=0$ is divisible by $n$

Hence $R$ is reflexive

$aRb\Rightarrow bRa$

Since $aRb$ if and only if $(a-b)$ is divisible by $n$ implies $bRa$ if and only if $(b-a)$ is divisible by $n$

$(b-a)=-(a-b)$ since $R$ is defined in $Z$

If $a-b$ is divisible by $n$

$-(a-b)$ is also divisible by $n$

Hence $R$ is symmetric

$aRb\in R$

$bRc\in R$

$a-b$ is divisible by $n$. and $b-c$ is divisible by $n$

But $(a-c)=(a-b)+(b-c)$

Hence $a-c$ is also divisible by $n$

$\therefore R$ is reflexive,symmetric and transitive

Hence $R$ is an equivalnce relation.