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I am having trouble constructing the sentences in this proof.

Its very simple, proof that [tex]A \cup \left( B \cap C \right) = \left( A \cup B \right) \cap \left( A \cup B \right) [/tex]

So basically I need to show that if [tex] x \in A \cup \left( B \cap C \right)[/tex] then [tex]x \in \left( A \cup B \right) \cap \left( A \cup B \right)[/tex]

Here is what I got:

If [tex] x \in A \cup \left( B \cap C \right)[/tex] then [tex] x \in A[/tex] or [tex] x \in \left( B \cap C \right)[/tex]. Which means that either [tex] x \in A[/tex], or [tex] x \in B[/tex] and [tex] x \in C[/tex]...

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First Question,

I feel like there is ambiguity here. "Either x in A, or x in B and x in C " can be interpreted two ways right? You could read it: [tex] x \in \left( A \cup B \right) \cap C [/tex] or you could read it as intended [tex] x \in A \cup \left( B \cap C \right)[/tex] How can I make the sentence clear that I want the latter?

Second question,

From here is it ok for me to make the jump to x in A or B, and x in A or C? It seems clear to me that this is the case, but I am not sure if something is left to be said before I make this claim.

Thanks for help!

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# Basic Set theory question

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