(a) Define a relation ~ on the set of real numbers by
Describe the equivalence class of .
(b) Prove that every equivalence class either consists entrely of rational numbers or else entirely of irrational numbers.
(c) On the set of all equivalence classes of with respect to ~ define the operation of addition as follows:
Prove that this operation is well defined, associative and that is a neutral element.
(d) Prove that if is a rational number than there exists a natural number such that
while if r is an irrational number than no such n exists.
(e) Is the following multiplication if elements of well defined?:
I'm not sure on the above. Thanks for any help. - Jason