Why do get that y belongs to this set?

[evinda]

Hello! (Nerd)

I am looking at the following exercise:

Prove that for all sets $A,B$, there are the sets:

$$\{ A \cap x: x \in B \} \text{ and } \{ A \cup x: x \in B \}$$
Show that:

• $A \cap \bigcup B=\bigcup \{ A \cap x: x \in B\} $
• for $B \neq \varnothing$, $A \cap \bigcup B=\bigcap \{ A \cap x: x \in B$

$$y: \exists x \in B (y=A \cap x) \rightarrow y \in \mathcal{P}(A \cap (\bigcup B))$$

How do we conclude that $y \in \mathcal{P}(A \cap (\bigcup B))$ ? (Thinking)

 

[Valkarie]

We start by observing that for all $y$, if there exists an $x \in B$ such that $y = A \cap x$, then $y$ is a subset of the set $A \cap (\bigcup B)$. This is because, for all $a \in y$, we have that $a \in A \cap x$, and since $x \in \bigcup B$, then $a \in A \cap (\bigcup B)$. Therefore, we can conclude that all elements of $y$ are also elements of $A \cap (\bigcup B)$, and hence $y \in \mathcal{P}(A \cap (\bigcup B))$.