Listing the elements of an equivalence class

Question

Say you have x = {1,2,3,4,5}, y ={2,5}, and c = {2,3} and the relation R: ARB iff AUY = BUY then asked to list all the equivalence classes of C

I had put down [2] and [3], with [2] = {3}, but I'm not too sure if this correct

Answer 1

You seem to have misunderstood "the equivalence class of $C$".

The equivalence class of $C$ is precisely the collection of sets $P$ such that $$P \cup Y = C \cup Y$$

Since $Y = \{ 2, 5\}$, that makes $C \cup Y = \{2,3,5\}$, so the equivalence class of $C$ is precisely every $P$ such that $$P \cup \{2,5 \} = \{2,3,5\}$$

Such a $P$ must contain the number $3$; then it may or may not contain any of the elements of $\{2, 5\}$.

So the equivalence class of $C$ is $$\{ \{3\}, \{2,3\}, \{3,5\}, \{2,3,5\}\}$$