# discrete mathematics – Construct an explicit bijection from [x], the equivalence class of x, to Q.

Define a relation $$sim$$ on the set of real numbers as follows: For $$x, y in mathbb{R}$$ : $$x sim y$$ if $$𝑥 −𝑦∈mathbb{Q}$$

I have proven that this is an equivalence class:

Reflexitivity

$$x – x = 0$$

$$0 in mathbb{Q}$$

so $$xsim x$$

Symmetry

$$x – y = a$$, where $$a in mathbb{Q}$$

$$y – x = -a$$, where $$-a in mathbb{Q}$$

so $$x sim y, y sim x$$

Transitive

$$x – y = a$$, $$a in mathbb{Q}$$,

$$y – z = b$$, $$z in mathbb{R}$$, $$b in mathbb{Q}$$

$$x – y + y – z = a + b$$

$$= x – z = a + b$$

$$= x – z = c$$, $$c in mathbb{Q}$$

Thus, $$x sim y$$ is an equivalence relation. However, I am having trouble finding the equivalence classes and most importantly, how to construct an explicit bijection from the equivalence class, $$[x]$$, to $$mathbb{Q}$$.