# How to prove the set distributive law for $n$ sets?

How do you show that if you have sets $B_1, B_2, \cdots ,B_n$ and a set $C$, then $$(B_1\cap B_2, \cap \cdots B_n)\cup C= (B_1\cup C)\cap(B_2\cup C) \cap \cdots \cap (B_n\cup C)\,?$$

Thanks

You have $$( B_1\cap B_2)\cup C = (B_1\cup C)\cap(B_2 \cup C).$$ Use an induction argument.

• Huh? this syntax makes no sense to me. Mar 5 '15 at 2:38
• What part of the syntax makes no sense? Mar 5 '15 at 2:44
• my bad it works now Mar 5 '15 at 2:58
• how would you start the inductive argument Mar 5 '15 at 3:11
• @tim29 Induction arguments have base cases and inductive steps. The base case is what ncmathsadist wrote in his answer. The inductive step involves assuming that your statement is true for $n=k$, and proving that it holds for $n=k+1$. The base case is the same as your equation if we set $n=2$. Mar 5 '15 at 5:24

You can prove the most general version without induction.

Corollary of (ii) \begin{align*} (B_1\cap B_2\cap\cdots \cap B_n)\cup C&= \left(\bigcap_{i=1}^{n}B_i\right)\cup C\\ &=\bigcap_{i=1}^n(B_i\cup C)\\ &=(B_1\cup C)\cap(B_2\cup C)\cap\cdots\cap(B_n\cup C) \end{align*}