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[SOLVED] B-C

dwsmith

Well-known member
Feb 1, 2012
1,673
$A\cap (B - C) = (A\cap B) - (A\cap C)$


For the identity, we will show $A\cap (B - C) \subseteq (A\cap B) - (A\cap C)$ and $A\cap (B - C) \supseteq (A\cap B) - (A\cap C)$.
Let $x\in A\cap (B - C)$.
Then $x\in A$ and $x\in B - C$.
So $x\in A$ and $x\in B$ and $x\notin B\cap C$.

Is this the right approach? I know $B-C = $ some other expression but I can't remember it.
 

Amer

Active member
Mar 1, 2012
275
[tex] B-C = B \cap C^c [/tex]
[tex]C^c [/tex] it is the complement of C, About you work it is right since x in A and in B so it is in the intersection and x it is not in C so it is not in the intersection of A and C

now the other direction
 

Plato

Well-known member
MHB Math Helper
Jan 27, 2012
196
$A\cap (B - C) = (A\cap B) - (A\cap C)$
$\begin{align*}(A\cap B)-(A\cap C)&= (A\cap B)\cap(A^c\cup C^c)\\&=(A\cap B\cap A^c)\cup (A\cap B\cap C^c) \\&=\emptyset \cup (A\cap B\cap C^c)\\&= (A\cap B\cap C^c)\\&=(A\cap B)-C\end{align*}$