TITLE: Trying to understand set theory? Subset vs Member?
QUESTION [10 upvotes]: 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).

REPLY [5 votes]: Looks good. Note that:
$$
Z \subseteq W \iff Z \in ℘(W)
$$
so statements $2$ and $3$ are equivalent.