Question

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

• A. reflexive
• B. symmetric
• C. transitive
• D. equivalence
• E. None of the above

Answer 1

The correct option is D equivalence

Let n be fixed positive integer

Def. Relation R on the set z of integers by $aRb\leftrightarrow n|a-b$

then what in Relation R?

Sol${}^n:-$ Let $aϵz$ then Relation of R on set z of integers by $aRb$

$\Rightarrow n|a-a$

$\Rightarrow n|0$ So $(a,a)ϵR$

So R in reflexive

Know for $a,b,ϵz$

Let $(a,b)ϵR\Rightarrow n|a-b$

$\Rightarrow n|-(b-a)$

$\Rightarrow n|-(b-a)\Rightarrow (b,a)ϵR$

So for $(a,b)ϵR$ we have $(b,a)ϵR$

So R is symmetric

now for $a,b,c,ϵz$

Let $(a,b)ϵR,(b,c)ϵR$

$\Rightarrow n|a-b$ and $n|b-c$

$\Rightarrow n|a-b+b-c$

$\Rightarrow n|a-c$

$\Rightarrow (a,c)ϵR$

Do for $(a,b)ϵR$ and $(b,c)ϵR$ we have $(a,c)ϵR$

So R in transitive

So R is reflexive, symmetric and transitive