1
$\begingroup$

Let $\{A_1,A_2\dots A_n\}$ be a mutually disjoint set of events. Let $\{B_1,B_2\dots B_n\}$ be another mutually disjoint set of events. I am trying to transform the expression $$C=\bigcap\limits_{i\in\{1,2,\dots,n\}}\left(A_i\cup B_i\right)$$ to something equivalent by somehow taking the union symbol out. I think that for $n=2$ it's $$C=(A_1\cap B_2)\cup(B_1\cap A_2)$$ but I cannot generalize it for any $n$.

$\endgroup$
  • $\begingroup$ @layman That's true but as I stated $\{A_1,A_2\dots A_n\}$ are mutually disjoint. This holds for $\{B_1,B_2\dots B_n\}$, as well. So $(A_1\cap A_2)=\emptyset$ and $(B_1\cap B_2)=\emptyset$ $\endgroup$ – mgus Nov 2 '17 at 19:44
1
$\begingroup$

I renamed your variables $A_{1,i}$ and $A_{2,i}$ $$C=\bigcap\limits_{i\in\{1,2,\dots,n\}}\left(A_{1,i}\cup A_{2,i}\right)$$

By generalized distributivity $$C=\bigcup\limits_{i_1\in\{1,2\}} \cdots \bigcup\limits_{i_n\in\{1,2\}} \left(\bigcap\limits_{j\in \{1,2,\cdots,n\}} A_{i_j,j} \right)$$

Which can be written as

$$C=\bigcup\limits_{(i_1,\cdots,i_n)\in\{1,2\}^n} \left(\bigcap\limits_{j\in \{1,2,\cdots,n\}} A_{i_j,j} \right)$$

This is a very ugly formula which means write your n brackets.

Choose one term either A or B from each bracket and take the intersection of those. Then take the union for every possible choice.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.