Question

Consider the following relations:
R = {(x, y)|x, y are real numbers and x = wy for some rational number w};

S = {(m/n, p/q)|m, n, p and are integers such that n, q ≠ 0 and qm = pn}.Then
(a) R is an equivalence relation but S is not an equivalence relation
(b) neither R nor S is an equivalence relation
(c) S is an equivalence relation but R is not an equivalence relation
(d) R and S both are equivalence relations

Answer 1

(c) : We have (x, x) ∈ R for w = 1 implying that R is reflexive.
For a ≠ 0, (a, 0) ∉ R for any w but (0, a) ∈ R. Thus R is not symmetric.
Hence R is not an equivalence relation.