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$$B \cap C \subseteq A \implies (C-A) \cap (B-A) = \varnothing.$$

I don't think this is true because B and neither C are necessarily a subset of A. Only B intercept C is a subset of A.

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    how come b intercept c is a subset of a implies that b and c are a subset of a? i can draw a diagram where only b intercept c is a subset of a and a = b intercept c.2012-10-30
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    The comment I made was when you had $B\cup C$ at the beginning, not $B\cap C$. Since it is not relevant to the changed question, I am deleting it.2012-10-30
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    Just so you're clear and for future reference: $B \cap C$ reads: "B intersect C"; i.e., the word to use is "intersect", not "intercept"...Also, to clear up any confusion: $(C - A)$ is often/usually denoted: "$(C \setminus A$)". (In Tex, that's (C \setminus A), enclosed in "$" signs here at MathSE.)2012-10-30

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