is $E(x),E(y) ↦ E(x+y)$ well defined? 
Let $x,y\in \mathbb R$ and $x\sim y \iff x-y\in \mathbb Z$. $E(x)$ is
  the equivalance class containing $x$.

a) Is $E((x),E(y)) ↦ E({x+y})$ well defined?
Where $\rightarrow$ means an operation
b) Is $(E(x),E(y)) ↦ E(xy)$?

If the notation $(E(x),E(y))$ is mean an operation, then $(E(x),E(y))=(E(z),E(w))$.
If the notation $(E(x),E(y))$ is mean an pair, then $(E(x),E(y))\neq(E(z),E(w))$. 
