(a) Define a relation ~ on the set $\displaystyle \mathbb{R}$ of real numbers by

$\displaystyle r$~$\displaystyle s \Leftrightarrow r-s \in \mathbb{Z}

$

Describe the equivalence class of $\displaystyle 0$.

(b) Prove that every equivalence class $\displaystyle [r]$ either consists entrely of rational numbers or else entirely of irrational numbers.

(c) On the set $\displaystyle C$ of all equivalence classes of $\displaystyle X$ with respect to ~ define the operation of addition as follows:

$\displaystyle [r] + [s] = [r+s]$

Prove that this operation is well defined, associative and that $\displaystyle [0]$ is a neutral element.

(d) Prove that if $\displaystyle r$ is a rational number than there exists a natural number $\displaystyle n$ such that

$\displaystyle \underbrace{[r] + ... + [r]}_n = [0] $

while if r is an irrational number than no such n exists.

(e) Is the following multiplication if elements of $\displaystyle C$ well defined?:

$\displaystyle [x][y]=[xy]

$

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