MHB Can Single Element Sets Be Subsets of Power Sets?

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The discussion centers on whether a single element set, such as {a}, can be a subset of the power set of the union of two sets A and B. It is established that if a is an element of A, then {a} is an element of the power set of A and consequently also of the power set of the union A ∪ B. The question arises about concluding that {a} is a subset of the power set of A ∪ B. Additionally, the participants explore how to demonstrate that the set containing {a} and {a, b} is an element of the power set of the power set of A ∪ B. The conversation encourages constructing examples to clarify these relationships.
evinda
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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 $\{ \{ a \}, \{ a, b \} \} \in \mathcal P \mathcal P (A \cup B)$ ? :confused:
 
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evinda said:
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?
 

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