Question

In this problem we consider another way to think about the rational numbers. Normally we would write fractions as p/q for p ∈ Z and q ∈ N. In this problem we represent fractions as ordered pairs. 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) if and only if ps = qr.

You should think about how this is related to the test that two fractions 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)? Now define 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?

Answer 1

For the 2nd part of the question, The set S has to be restricted under the condition that, two elements of S will be equal iff they belong to same class or they are related to each other. Otherwise '<' will never be a partial order.

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