Question

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

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

Answer 1

Answer: (D)

equivalence

Reflexive: For any $a\in N$, we have $a-a=0=0\times n\Rightarrow -a$ is divisible by $n\Rightarrow (a,a)\in R$

Thus $(a,a)\in R$ for all $a\in Z$. So, $R$ is reflexive.

Symmetry: Let $(a,b)\in R$. Then,

$\Rightarrow (a,b)\in R\Rightarrow (a-b)$ is divisible by $n$.

$\Rightarrow (a-b)=np$ for some $p\in Z$

$\Rightarrow b-a=n(-p)$

$\Rightarrow b-a$ is divisible by $n$

Thus, $(a,b)\in R\Rightarrow (b,a)\in R$ for all $a,b\in Z$

So, $R$ is symmetric on $Z$.

Transitive: Let $a,b,c\in Z$ such that $(a,b)\in R$ and $(b,c)\in R$. Then $(a,b)\in R\Rightarrow (a-b)$ is divisible by $n$.

$\Rightarrow a-b=np$ for some $p\in Z$

$(b,c)\in R\Rightarrow (b-c)$ is divisible by $n$.

$\Rightarrow b-c=nq$ forsome $q\in Z$

therefore,$(a,b)\in R$ and $b-c\in R$

$\Rightarrow a-b=npb-c=nq$

$\Rightarrow (a-b)+(b-c)=np+nq$

$a-c=n(p+q)$

$\Rightarrow a-cisdivisibleby$n$.

$\Rightarrow (a-c)\in R$

Thus, $(a,b)\in R$ and $(b,c)\in R$

$\Rightarrow (a,c)\in R$ for all $a,b,c\in Z$

So, $R$ is transitive relation on $Z$.

Thus, $R$ being reflexive, symmetric and transitive, is an equivalence relation