Prove that if $A_1, A_2, \ldots , A_n$ and $B$ are sets, then $$(A_1 − B) \cap (A_2 − B) \cap \cdots \cap (A_n − B) = (A_1 \cap A_2 \cap \cdots \cap A_n) − B.$$
Prove that $\bigcap\limits_{i = 1}^n {\left( {{A_i} - B} \right)} = \bigcap\limits_{i = 1}^n {{A_i}} - B$
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elementary-set-theory
discrete-mathematics
induction
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1Where did this problem come from? What have you tried? It matches [this Yahoo! Answers question](http://answers.yahoo.com/question/index?qid=20111116154729AAZU0xo) word for word. – 2012-10-28