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I'm having trouble wrapping my head around this question. Here is what I have for my proof so far:

The statement is true. Assume $A, B, C$ are sets and $A - (B\cap C) =\varnothing$. Now we prove $A - B =\varnothing$ and $A - C = \varnothing$.

I've tried several things from this point, but none of them seem to work. I've tried saying that since $A - (B\cap C) =\varnothing$, $A$ is a subset of ($B\cap C$). Which makes sense, except the set definitions don't allow me to conclude this.

What am I able to derive from my assumption? Also, I want to say this: To prove this we need $A =\varnothing$, or we need to know that $A$ is a subset of both $B$ and $C$. Is this true? Do the set definitions allow us to conclude that $A - B =\varnothing$ if $A$ is a subset of $B$?

Is there something I'm not seeing or am I thinking about the problem wrong?

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1 Answers1

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Observe that $$(A \setminus B) \cup (A \setminus C) = A \setminus (B \cap C) = \varnothing$$ and recall that $X \cup Y = \varnothing$ implies that $X = \varnothing$ and $Y = \varnothing$.

azif00
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  • But how did you prove that (A ∖ B) ∪ (A ∖ C) = A ∖ (B ∩ C) = ∅? I don't quite understand how to apply the rules and definitions of sets when dealing with identities. Like how do you apply the set rules to rearrange identities like the one in my assumption? –  Oct 17 '20 at 01:29