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

Both reflexive and symmetric
Reflexive but not symmetric
Not reflexive but symmetric
Neither reflexive nor symmetric

Solution:

The key trick here is to realize that the relation is of the form :

{ordered pair, ordered pair} and not simply ordered pair.

Ok, so for reflexive

$\forall\_{a,b}\, if((a,b),(a,b)) \in \mathrel{R} \rightarrow \text{reflexive}$

$((a,b),(a,b))∈R↔(a−b=b−a)$ (not possible for any postive integers b and a)

But that is a contradiction hence it is not reflexive.

Now, for symmetric

$((a,b),(c,d))∈R→((c,d),(a,b))∈R$
$((a,b),(c,d))∈R→(a−d=b−c)$
$((c,d),(a,b))∈R$

$∵(c−b=d−a)↔(d−a=c−b)↔(−(a−d)=−(b−c))↔(a−d=b−c)$

This is the solution given in from trustworthy source of technical examiniation.

But my doubt is If we are saying that
$((a,b),(c,d))∈R→((c,d),(a,b))∈R$ to be symmetric right?

Why are we checking symmetric relation on ordered pair and reflexive relation on only two pairs $((a,b),(a,b))∈R$

I know Reflexive relation is said to be reflexive if $xRx \forall\_{x}\ $ in $R$