In this problem we consider another way to think about therational numbers. Normally we would write fractions as p/q for p ?Z and q ? N. In this problem we represent fractions as orderedpairs. So let S = {(p, q)|p ? Z and q ? N}.
For ordered pairs (p, q) and (r, s) in S define (p, q)R(r, s) ifand only if ps = qr.
You should think about how this is related to the test that twofractions are equal.
a. Prove that R is an equivalence relation on S.
b. What is the equivalence class that contains (0, 1)?
c. What is the equivalence class that contains (2, 1)? Nowdefine a partial order (p, q) ? (r, s) for (p, q) ? S and (r, s) ?S. Answer each of the following question and prove your result.
d. Is this a reflexive relation?
e. Is it symmetric?
f. Is it antisymmetric?
g. Is this a transitive relation?
