Let $S$ be the set of all subsets of $\mathbb{Z}$. Define a relation $\sim$ on $S$ via $T \sim U$ if and only if $T \setminus U $ and $U\setminus T$ are both finite. Show that $\sim$ is an equivalence relation and describe $[\{1, 2, 3\}]$ and $[\{. . . ,−4,−2, 0, 2, 4, . . .\}]$.
My attempt.
Reflixivity: Since $U\setminus U=\emptyset $, we have that $U\sim U$.
Symmetry: If $T \sim U$ , then $T \setminus U $ and $U\setminus T$ are both finite. Thus $U \sim T$
Trnsitivity: (Here my problem begins) If $T \sim U$ and $U \sim V$, we have that $T \setminus U,\; U \setminus T,\; U \setminus V $ and $V \setminus U $ are all finite. How can I relate $T$ to $V$?
On the other hand, I think that the elements of $[\{1, 2, 3\}]$ are all elements of finite cardinality, since their differences will always be finite. Whereas for the elements of $[\{. . . ,−4,−2, 0, 2, 4, . . .\}]$ I have no idea.
You have to prove that $T\setminus V$ and $V\setminus T$ are both finite given that $T\setminus U; etc.$ are all finite.
Let $x \in T\setminus V$ then $x \in T$ . Now either $x \in U$ or $x\not \in U$. If $x \in U$ we have $x \not \in V$ so $x\in U\setminus V$. But if $x \not \in U$ we have $x \in T \setminus U$. So $x \in U \setminus V$ or $x \in T\setminus U$. So $x \in (U\setminus V)\cup (T\setminus U)$.
So $T\setminus V \subseteq (U\setminus V)\cup (T\setminus U)$ which is the union of two finite sets.
Now... you should have a theorem that the union of two finite subsets is finite so $(U\setminus V)\cup (T\setminus U)$ is finite.
And you should have a theorem that a subset of a finite set is finite. So $T\setminus V$ is finite.
ANd proving $V\setminus T$ is finite is identically proven.