Does the empty set count as an element?

Question

We're asked the number of elements in a power set and I for {} is the number of elements 1 or 0?

Answer 1

No, an empty set is a subset of every set, not necessarily an element. However, the empty set is an element of the power set of any set.

Notice: $X\subset Y\iff \forall x(x\in X \implies x \in Y)$ which is vacously true for any arbitrary set and $\emptyset$ since $\lnot \exists x(x\in \emptyset)$. However, to say $\forall A(\emptyset \in A)$ implies for a Set $B={a,b,c}$ its $|B|=4$, which is obviously false.

Hope that answers your question.