MHB Why Prove \( x = y \) in \( \mathcal{P} \cup A \)?

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The discussion focuses on proving that for any set \( A \), \( A \subset \mathcal{P} \cup A \). It clarifies that if \( x \in A \), then \( x \) must also be in \( \mathcal{P} \cup A \) by demonstrating that every element \( z \) in \( x \) is part of the union of \( A \). Participants also discuss the distinction between the binary operator \( \cup \) and the unary operator \( \bigcup \), emphasizing the correct notation in set theory. The conversation further explores the equality \( A = \mathcal{P}(\bigcup A) \), concluding that it holds only under specific conditions, namely when \( A \) is the power set of some set \( B \). The thread highlights the importance of understanding set operations and their implications in proofs.
evinda
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Hi! (Smile)

I want to prove that for each set $A$:
$$A \subset \mathcal P \cup A$$

According to my notes, we prove it like that:

Let $x \in A$. We want to show that $x \in \mathcal P \cup A$, so, that: $\exists y \in \mathcal P \cup A$, such that $x=y$.
It suffices to show that if $z \in x$, then $z \in \cup A$, that stands, since $x \in A$.

Could you explain me why we want to show that :
$$\exists y \in \mathcal P \cup A, \text{ so that } x=y$$
?
At the beginning, we suppose that $x \in A$ and we want to prove that $x \in \mathcal P \cup A$, so we want to prove that $x \subset \cup A$.
I haven't understood what we could do to prove this.. (Worried)
 
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evinda said:
I want to prove that for each set $A$:
$$A \subset \mathcal P \cup A$$
Do you mean $\mathcal{P}\left(\bigcup A\right)$? In LaTeX, \cup is a binary operator, while \bigcup is a prefix unary operator.

evinda said:
Could you explain me why we want to show that :
$$\exists y \in \mathcal P \cup A, \text{ so that } x=y$$?
This statement is equivalent to $x\in\mathcal P( \bigcup A)$, and I don't see how it simplifies it.

evinda said:
At the beginning, we suppose that $x \in A$ and we want to prove that $x \in \mathcal P \cup A$, so we want to prove that $x \subset \cup A$.
Yes. The latter statement means $\forall z\;z\in x\to z\in\bigcup A$. But this follows from the definition of $\bigcup A$.
 
Evgeny.Makarov said:
Do you mean $\mathcal{P}\left(\bigcup A\right)$? In LaTeX, \cup is a binary operator, while \bigcup is a prefix unary operator.

Yes, that's what I meant. I will take it into consideration! (Smile)

Evgeny.Makarov said:
Yes. The latter statement means $\forall z\;z\in x\to z\in\bigcup A$. But this follows from the definition of $\bigcup A$.

A ok! I got it! (Nod)

And does the equality $A= \mathcal{P}\left(\bigcup A\right) $ also stand? (Thinking)
 
evinda said:
And does the equality $A= \mathcal{P}\left(\bigcup A\right) $ also stand?
No. For example, when $A=\{\{1,2\},\{3,4\}\}$, then $\bigcup A=\{1,2,3,4\}$ and $\mathcal{P}(\bigcup A)$ contains all 16 subsets of $\{1,2,3,4\}$. Of those 16, $A$ contains only two.
 
Evgeny.Makarov said:
No. For example, when $A=\{\{1,2\},\{3,4\}\}$, then $\bigcup A=\{1,2,3,4\}$ and $\mathcal{P}(\bigcup A)$ contains all 16 subsets of $\{1,2,3,4\}$. Of those 16, $A$ contains only two.

I understand! (Nod) Does the equality never hold or are there cases, where it stands that $A=\mathcal{P}(\bigcup A)$ ? (Thinking)
 
evinda said:
Does the equality never hold or are there cases, where it stands that $A=\mathcal{P}(\bigcup A)$ ?
This happens iff $A=\mathcal{P}(B)$ for some $B$.
 
Evgeny.Makarov said:
This happens iff $A=\mathcal{P}(B)$ for some $B$.

How did you conclude that it happens iff $A=\mathcal{P}(B)$ for some $B$?

And how could we prove this? (Thinking)
 

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