Trying to understand set theory? Subset vs Member?

So I'm learning set theory and wanted to know if I'm understanding some of the basics correctly. I have a feeling 1 and 2 are correct, but I'm not sure about 3 and 4.

Z = {1,3}
W = {1,2,3,4}

1) Is Z ∈ W? No, Z is not a member or W, because Z is the set {1,3}, while the members of W are 1,2,3,4.

2) Is Z ⊆ W? Yes, because all elements in the set Z (1 and 3) are also present in the set W.

3)Is Z ∈ ℘(W)? Yes, because ℘(W) is a set of sets, Z is a member of ℘(W) because {1, 3} is one of those sets.

4)Is Z ⊆ ℘(W)? No, Z is not a subset of the powerset of W because the elements in set Z (1 and 3) are not present in the set ℘(W).

Solution 1:

This is all correct, except for the statement in 3) that "Z is a power set of ℘(W)" which should read "Z is an element of ℘(W)". Note that ℘(W) is called the power set of W.

Solution 2:

Looks good. Note that: $$ Z \subseteq W \iff Z \in ℘(W) $$ so statements $2$ and $3$ are equivalent.