Relations and Equivalence - numbers are related if they have the same floor $S$ is defined on $\mathbb{Q}$ by $xSy$ if and only if $⌊x⌋=⌊y⌋$ (Note that$⌊q⌋$is defined to be the largest integer less than or equal to q. You can think of it as “$q$ rounded down”.)
We've been asked to find the relations of this. So far I have figured out that these are Reflexive, Symmetric and Transitive making it an equivalence relation. However we are required to identify the class. Just unsure how to identify that
Thanks
 A: Take some number $q\in \mathbb{Q}$. We want to figure out what numbers are related to $q$ through the relation $S$.
To do this, you could begin with thinking of an example. Take for instance $q=\frac{7}{3}=2.33333...$. Rounded down, this becomes $\lfloor q\rfloor = 2$. The key point here is that all other numbers that rounded down become $2$ will exactly be the numbers that are related to $q$ (for instance, since $\lfloor 2.1\rfloor = 2$, we have that $2.1$ is related to $q$). 
And remember, the equivalence class of $q$ is exactly the set of numbers that are related to $q$.
A: If $f:X\rightarrow Y$ is a function then it induces an equivalence relation $\sim$ on $X$ by: $$a\sim b\iff f(a)=f(b)$$
Denoting the equivalence class of $a\in X$ by $[a]$ we come to: $$[a]=\{x\in X\mid x\sim a\}=\{x\in X\mid f(x)=f(a)\}$$
The set of equivalence classes is: $$\{f^{-1}(\{y\})\mid y\in\text{im}f\}$$
In your case: $$\text{im}f=\mathbb Z\text{ and }f^{-1}(\{n\})=[n,n+1)\cap\mathbb Q\text{ for }n\in\mathbb Z$$
