Is this website correct about p(∅)?

Question

I am learning about power sets.

I know that p(∅) = {∅} or {{}}.

That is a little confusing to me, so I used an online calculator (https://www.mathsisfun.com/sets/power-set-maker.html). After putting in ∅ they tell me that power set of ∅ is {{}, {∅}}. Now I am more confused.

Is {{}, {∅}} the same as {∅}?

Answer 1

I will first mention that if we're given a set A with cardinality |A|, then the cardinality of the power set of A is given as follows: |P(A)| = 2^|A|.

Taking A=∅, we know that the power set of ∅ should contain 2^|∅|=2⁰=1 elements. Furthermore, we know that P(A) is the set containing all possible subsets of A; thus if A=∅, then the only subset ∅ has is ∅; therefore, P(∅)={∅}, which is a singleton (one-element) set where the only element is the empty set.

On the other hand, if you chose A = {∅}, then A is a single-element set (singleton) and so it must have 2^|{∅}| = 2¹=2 subsets. We furthermore note that the only subsets of a singleton set are the empty set and the singleton set itself; hence, when A={∅}, we find that P(A) = {∅,{∅}}.

Note here that {∅,{∅}} and {∅} are not the same set; the first set is a two-element set with elements ∅ and {∅}; the second set is a single-element set where the sole element is the empty set ∅. So even if the elements of these sets are empty sets (or sets of empty sets), the overall sets containing them aren't empty (confusing, I know).

To generate the correct subset using the link you provide, you want to generate a power set for a blank input; the input is already assuming that whatever you're providing as input are elements in the set; so if the set is empty don't type anything; then you'll see it will generate the correct element in the power set which would be {}=∅. Typing in {} would assume that the set you're finding the power set for is {{}} = {∅} ≠∅; hence why it gave you the set {∅,{∅}} as an answer for an input of {}.

This can be confusing at times, but I hope this clarifies some things about the power set of an empty set!