Exploring the Powerset of Sets: How to Find All Elements Using Simple Identities

[evinda]

Hello! (Wave)

I want to find the powerset $\mathcal{P} \mathcal{P} \{ \varnothing, \{ \varnothing \} \}$, which has $16$ elements.

I found that $\mathcal{P} \{ \varnothing, \{ \varnothing \} \}=\{ \varnothing, \{ \{ \varnothing \} \}, \{ \varnothing \}, \{ \varnothing, \{ \varnothing \} \} \}$.

Using the identities:

• The empty set $\varnothing$ is a subset of each set.
  $$$$
• If $A$ is a set, then $A \subset A \rightarrow A \in \mathcal{P} A$.
  $$$$
• If $A$ is a set and $x \in A$, then:
  $$\{ x \} \subset A \rightarrow \{ x \} \in \mathcal{P} A$$

I found:

$$\mathcal{P} \{ \varnothing, \{ \{ \varnothing \} \}, \{ \varnothing \}, \{ \varnothing, \{ \varnothing \} \} \}=\{ \varnothing, \{ \varnothing \}, \{ \{ \{ \varnothing \} \} \}, \{ \{ \varnothing \} \}, \{ \{ \varnothing, \{ \varnothing \} \} \}, \{ \varnothing, \{ \{ \varnothing \} \}, \{ \varnothing \}, \{ \varnothing, \{ \varnothing \} \} \}$$

Which other identities could I use to find the other elements of the powerset? (Thinking)

 

[Evgeny.Makarov]

Which other identities could I use to find the other elements of the powerset?

Why do you need any identities?
\[
\mathcal{P}\{1,2,3,4\}=\{\varnothing,\{1\},\{2\},\{3\},\{4\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\},\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},\{1,2,3,4\}\}
\]
Replace in this identity 1, 2, 3, 4 with $\varnothing$, $\{ \varnothing \}$, $\{ \{ \varnothing \} \}$, $\{ \varnothing, \{ \varnothing \} \}$, respectively.

 

[evinda]

Why do you need any identities?
\[
\mathcal{P}\{1,2,3,4\}=\{\varnothing,\{1\},\{2\},\{3\},\{4\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\},\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},\{1,2,3,4\}\}
\]
Replace in this identity 1, 2, 3, 4 with $\varnothing$, $\{ \varnothing \}$, $\{ \{ \varnothing \} \}$, $\{ \varnothing, \{ \varnothing \} \}$, respectively.

I understand... (Nod) Is the number of different arrangements of $j, 1 \leq j \leq 4$ numbers equal to $\binom{4}{j}$ ? (Thinking)