Set difference is finite - transitive relation?

Question

Let $A=P(\mathbb N)$. The relation $E$ is defined: $(X,Y) \in E$ iff $X \setminus Y$ and $Y \setminus X$ are finite.

I was given to prove this is an equivalence relation, however I had troubles proving the transitivity. Any hints?

Answer 1

Let $(X,Y) \in E$ & $(Y,Z)∈E$. we prove $(X,Z) \in E$ (in one direction, the other will be similar):
if $c\in Z \setminus X$ then we have two possibility:
1) $c \in Y$ then $c\in Y\setminus X$, which is finite
2) $c\notin Y$ then $c\in Z\setminus Y$, which is finite

Answer 2

Hint: Try to prove $X\setminus Z \subseteq (X\setminus Y) \cup (Y\setminus Z)$ first.