Use the element method to prove the following statement.

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Use the element method to prove the following statement. For all sets A, B, and C, if B ∩ C ⊆ A then (C − A) ∩ (B − A) = ∅.

I have no clue on how to do this. Please show/guide me

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1 Answer

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Assume $x \in (B-A) \cap (C-A)$. This means precisely that $x \in B \cap C$ and $x \not\in A$. But we know that $B \cap C \subset A$, so $x \not\in A$ implies $x \not\in B \cap C$, a contradiction. Since we assumed $x \in (B-A) \cap (C-A)$ and derived a contradiction this means $(B-A) \cap (C-A) = \varnothing$.

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