Function and Equivalence Classes question

Question

I have the following question:

Question: For the set $\theta=\{(X,|X|):X\subseteq\mathbb{Z}_5\}$, what is its domain and range?

The Domain stated in the solution from the textbook is: $\mathcal{P}(\mathbb{Z}_5)$ as in the Power set of $\mathbb{Z}_5$. But is it alright if I write the domain in terms of equivalence classes: $\text{Domain}=\{[0],[1],[2],[3],[4]\}$, where $[0],\ldots,[4]$ represents the equivalence classes.

The Range stated in the solution from the textbook is: $\{0,1,2,3,4,5\}$. I don't understand this part, because as stated, the set "$\theta$" where the range is a list of $|X|$ values, so if $X=[0]$, then $[0]=\{\ldots,-10,-5,0,5,10,\ldots\}$ since $\mathbb{Z}_5$ means "$\equiv\,(\mathrm{mod}\, 5)$", so $[0]$ means $x\equiv0\,(\mathrm{mod}\, 5)$. So $|X|$ when $X=[0]$ is the carnality of a countable infinite set.... which don't make sense.

So why is the Range of the set "$\theta$" $=\{0,1,2,3,4,5\}$?

Thanks

Answer 1

Even though each element of $Z_5$ is an infinite equivalence class, the set $Z_5$ itself is finite. The domain of $\theta$ is not $\{[0], \dots [4]\}$; rather, it's all the subsets, so the domain includes $\{[0]\}$, and $\{[1]\}$, and $\{[0],[1]\}$, and $\{\}$, as 4 examples. The domain has 32 elements. The range is in turn the sizes of these subsets. For instance, the maximum of the function is the input $\{ [0], [1], [2], [3], [4] \}$, which has output 5. The input $\{ [0], [1], [3], [4] \}$ is also in the domain, and has output 4. I think you're confusing the elements of $Z_5$ with the subsets of $Z_5$.