$S$ 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
On
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: $$\text{im}f=\mathbb Z\text{ and }f^{-1}(\{n\})=[n,n+1)\cap\mathbb Q\text{ for }n\in\mathbb Z$$
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 $q$.