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?
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Let R be a relation on ordered pairs of positive integers such that ((p,q),(r,s)) R iff p-s=q-r
Check reflexive:
For reflexive, take (p,q)=(r,s)
Then condition becomes
p-q=q-p
2p=2q
p=q
This is not true for every ordered pair, so R is not reflexive.
Check symmetric:
Given p-s=q-r
Rearrange: p+r=q+s
Now for symmetry we need
r-q=s-p
This also gives r+p=s+q, same condition.
So R is symmetric.