Define $\equiv$ and $\sim$ on $\mathbb{R}$ by $x\equiv y$ if $x-y \in \mathbb{Z}$ and by $x\sim y$ if $x-y\in \mathbb{Q}$.
a) Show that $\equiv$ and $\sim$ are equivalences.
b) Show that $\alpha:\mathbb{R}_\equiv \rightarrow \mathbb{R}_\sim$ is well defined and onto if $\alpha([x]_\equiv)=[x]_\sim$. Is the mapping $\alpha$ one to one?
For a I know that to show $\equiv$ and $\sim$ are equivalences I must show that each is reflexive, symmetric and transitive. I have done so with things like: $(n,m)\equiv (n_1,m_1)$ if $n-m = n_1-m_1$, but I don't know where to begin with these.
It's the same story for part b, I've shown well defined, onto, and one to one before, but not sure how to in this case.
a) Reflexive : does $a-a\in \mathbb{Z}$ (resp. $\mathbb{Q}$)?
Symmetric : assume that $b-a\in \mathbb{Z}$ (resp. $\mathbb{Q}$), does $a-b\in \mathbb{Z}$ (resp. $\mathbb{Q}$)? (Hint : if a number $z$ is in $\mathbb{Z}$ resp. $\mathbb{Q}$ then its opposite $-z$ is in $\mathbb{Z}$ resp. $\mathbb{Q}$).
Transitive : assume that $b-a\in \mathbb{Z}$ (resp. $\mathbb{Q}$) and $c-b\in \mathbb{Z}$ (resp. $\mathbb{Q}$), does $a-c\in \mathbb{Z}$ (resp. $\mathbb{Q}$)? (Hint : if two numbers $z_1$ and $z_2$ are in $\mathbb{Z}$ resp. $\mathbb{Q}$ then their sum $z_1+z_2$ is also in $\mathbb{Z}$ resp. $\mathbb{Q}$).
2) Are you sure this is one-to-one ? Take $[0]$ and $[0.5]$, show that they are not the same class mod $\mathbb{Z}$ but $\alpha([0])=\alpha([0.5])$...