Power Sets of $\{a,b\}$ and $\{\emptyset,\{\emptyset\}\}$

A power set of a set $S$ (written $\mathcal{P}(S)$) is the collection of every subset of $S$, including the empty set $\emptyset$ and the set $S$ itself. For a finite set with $n$ elements, the number of subsets—and hence the size of the power set—is $2^n$.2

For computational thinking, you can view “finding the power set” as choosing, for each element of $S$, whether it is included in a subset. That yields $2^n$ distinct subsets.

Let’s apply the definition carefully to two specific sets, emphasizing what counts as an element.

(i) $S=\{a,b\}$

Here $S$ has $n=2$ elements, so $\lvert \mathcal{P}(S)\rvert = 2^2=4$. The subsets are obtained by including/excluding each of $a$ and $b$.

(ii) $T=\{\emptyset,\{\emptyset\}\}$

This set has $n=2$ elements as well, but note the elements are:

• one element is $\emptyset$,
• the other element is $\{\emptyset\}$ (a set whose only element is $\emptyset$).

So $\lvert \mathcal{P}(T)\rvert=4$. The subsets of $T$ are formed by choosing whether to include the element $\emptyset$, whether to include the element $\{\emptyset\}$, etc.—always treating them as atomic elements when forming subsets.

(i) Power set of $\{a,b\}$

All subsets of $\{a,b\}$ are:

$\mathcal{P}(\{a,b\})=\{\emptyset,\{a\},\{b\},\{a,b\}\}.$

You can verify the count: $2^2=4$ subsets.

(ii) Power set of $\{\emptyset,\{\emptyset\}\}$

First, write the elements of the base set $T=\{\emptyset,\{\emptyset\}\}$ explicitly:

• Element 1: $\emptyset$
• Element 2: $\{\emptyset\}$

Now list all subsets of $T$ by include/exclude of those two elements:

Interpretation of each subset:

• $\emptyset$ contains nothing.
• $\{\emptyset\}$ contains exactly the element $\emptyset$.
• $\{\{\emptyset\}\}$ contains exactly the element $\{\emptyset\}$.
• $\{\emptyset,\{\emptyset\}\}$ contains both elements.

Again, there are $2^2=4$ subsets.