Can Single Element Sets Be Subsets of Power Sets?

[evinda]

Hey! (Wave)

Knowing that $A,B$ are sets, and:

$$\text{ If } a \in A, \text{ then } \{ a \} \subset A \rightarrow \{ a \} \in \mathcal P A \rightarrow \{ a \} \in \mathcal P (A \cup B)$$

from this: $\{ a \} \in \mathcal P (A \cup B)$, can we conclude that:
$$\{ a \} \subset \mathcal P (A \cup B)$$

? (Thinking)

Also, when we have $\{ a \} \in \mathcal P( A \cup B)$ and $\{ a, b \} \in \mathcal P (A \cup B)$, how do we conclude that $\{ \{ a \}, \{ a, b \} \} \in \mathcal P \mathcal P (A \cup B)$ ?

 

[Siron]

Hey! (Wave)

Knowing that $A,B$ are sets, and:

$$\text{ If } a \in A, \text{ then } \{ a \} \subset A \rightarrow \{ a \} \in \mathcal P A \rightarrow \{ a \} \in \mathcal P (A \cup B)$$

from this: $\{ a \} \in \mathcal P (A \cup B)$, can we conclude that:
$$\{ a \} \subset \mathcal P (A \cup B)$$

? (Thinking)

Construct a simple example for yourself. For example, let $A =\{a_1\}$ and $B=\{b_1,b_2\}$. Take a look at $\mathcal{P}(A), \mathcal{P}(B)$ and $\mathcal{P}(A \cup B)$.

What do you think?