Question

Let $R$ be a relation on the set of ordered pairs of positive integers such that $\left(\right(p,q),(r,s\left)\right)ϵR$ if and only if $p-s=q-r$. Which one of the following is true about $R$?

• A. Both reflexive and symmetric
• B. Reflexive but not symmetric
• C. Not reflexive but symmetric
• D. Neither reflexive nor symmetric

Answer 1

The correct option is C Not reflexive but symmetric

$R=\left\{\right(p,q),(r,s)|p-s=q-r\}$
(i) Check reflexive property
$\forall (p,q)ϵZ^{+}\times Z^{+}\left(\right(p,q),(p,q\left)\right)ϵR$
is true iff $p-q=q-p$ which is false.
So relation $R$ is not reflexive.
(ii) Check symmetry property
If $\left(\right(p,q),(r,s\left)\right)ϵR$ then $\left(\right(r,s),(p,q\left)\right)ϵR$
$\left(\right(p,q),(r,s\left)\right)ϵR\Rightarrow p-s=q-r$
$\left(\right(r,s),(p,q\left)\right)ϵR\Rightarrow r-q=s-p$
If $p-s=q-r$ is true, then $r-q=s-p$ is also true by rearranging the equation.
$\therefore R$ is symmetric.