Question

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?
1. Both reflexive and symmetric
2. Reflexive but not symmetric
3. Not reflexive but symmetric
4. Neither reflexive nor symmetric

Answer 1

Correct Answer - Option 3 : Not reflexive but symmetric

Given: ((p, q), (r, s)) ∈ R

Only if p – s = q – r

Consider,

If ((p,q), (p,q)) ∈ R, then, p – q = q – p

It is not true for any positive integer p and q.

So, relation is not reflexive.

For symmetric: if ((p,q),(r,s)) ∈ R then also, ((r,s), (p,q)) ∈ R

Means if p – s = q – r then r – q = s – p

Both are equal, so relation is symmetric.