(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?:

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I'm not sure on the above. Thanks for any help. - Jason