Needing some help with Discrete

[Angrboda]

Currently studying for a test and noticed this problem


2.1.23. How many elements does each of these sets have where a and b are distinct elements?

A. P({a, b, {a, b}})
B. P({∅, a, {a}, {{a}}})
C. P(P(∅))

For whatever reason this was not covered in class (It's not in my notes and i take notes in that class religiously) Was wondering if anyone knew how the hell do this, the book we were assigned does a poor job of explaining it, or anything for that matter.

Writing A = {1, 2, 3, 4 } means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example {1, 2}, are subsets of A.

Sets can themselves be elements. For example consider the set B = {1, 2, {3, 4}}. The elements of B are not 1, 2, 3, and 4. Rather, there are only three elements of B, namely the numbers 1 and 2, and the set {3, 4}.

The elements of a set can be anything. For example, C = { red, green, blue }, is the set whose elements are the colors red, green and blue.

A. P({a, b, {a, b}})
B. P({∅, a, {a}, {{a}}})
C. P(P(∅))


It's been too long, so don't take my word for it, but...

A. one element (the set {a,b,{a,b}}), or three, depending.
B. one element or four, again depending.
C. Not sure. I don't know if the empty set counts as an element; my guess would be, yes, it does.