Question

A relation R is defined on the real numbers by aRb if a-b is an integer. Prove that R is an equivalence relation and determine the equivalence class of 1/2

Answer 1

You could try this type of argument, but I'm questioning my own correctness...

if a and b and c must be real, then if d, e, q, r and n are integers:

\[\frac{d}{e} - \frac{q}{r} = n\]

where a = c/d and b = q/r, which implies:

dr - qe = ner .... which are also all integers.

Then the following is true for all integer pairs Int:Int,

if a = d:r and b = q:e and ( c = d:r or q:e not equal with a or b respectively )

(a, a), (b, b), (c, c) are all members of R (reflexive)
(a, b), (b,a) [where a or b are interchangable with c] are both members of R (symmetric)
(a, b) , (b, c) and (a,c) are all members of R (transitive)

Answer 2

anyone have any other ideas?