Power Set: Membership versus Inclusion

The power set $P(A)$ (also written $2^{A}$) is the set of all subsets of $A$. This single fact collapses two relations that look different on the page into one:

$S \in P(A) \iff S \subseteq A.$

The symbol $\in$ on the left (membership in $P(A)$) is the same test as $\subseteq$ on the right (inclusion in $A$). So a question that asks "is $S$ an element of $P(A)$?" is really asking "is every element of $S$ also an element of $A$?"

Two relations, kept straight

Two facts always hold, because $\phi \subseteq A$ and $A \subseteq A$:

$\phi \in P(A), \qquad A \in P(A).$

And if $A$ is finite with $|A| = n$, then $|P(A)| = 2^{n}$.

Counting elements is the first move

Membership checks go wrong when you miscount the elements of the outer set, especially with nested empty sets. These three are distinct:

• $\phi$ — the empty set, $0$ elements;
• $\{\phi\}$ — a set whose one element is $\phi$, so $1$ element;
• $\{\{\phi\}\}$ — a set whose one element is $\{\phi\}$, so $1$ element.

Braces matter: peel one layer at a time.

Worked mini-example

Let $A = \{\, \phi,\ \{\phi\},\ \{\{\phi\}\} \,\}$ — a set with exactly three elements. Test two candidates for membership in $P(A)$.

Candidate $\{\{\phi\}\}$. This is the singleton whose only element is $\{\phi\}$. Is $\{\phi\} \in A$? Yes — it is the second listed element. Every element of the candidate lies in $A$, so $\{\{\phi\}\} \subseteq A$, hence $\{\{\phi\}\} \in P(A)$. ✓

Candidate $\{\, \phi,\ \{\phi\} \,\}$. Both $\phi \in A$ and $\{\phi\} \in A$, so this is a subset of $A$ and therefore an element of $P(A)$. ✓

Notice the trap: $\{\{\phi\}\}$ is a member of $P(A)$ even though, as a set, it is not one of the three listed elements of $A$ — what matters is that its contents sit inside $A$.

Pitfall: "$S \in P(A)$" never means "$S$ is one of the listed elements of $A$." It means "$S \subseteq A$," i.e. every element of $S$ belongs to $A$. Before testing anything, count the distinct elements of $A$ by peeling braces one layer at a time — nested empty sets like $\phi$, $\{\phi\}$, $\{\{\phi\}\}$ are three different objects.