# Distributivity of intersections across unions

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Created for use with the ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions, the theorem is easy and routine, at least in the finite cases

## Contents

## Statement

- [ilmath]A\cap(B\cup C)=(A\cap B)\cup(A\cap C)[/ilmath]
- [ilmath]A\cap(\bigcup_{i=1}^n B_i)=\bigcup_{i=1}^n(A\cap B_i)[/ilmath] -
**Easy to do, use induction**

## Proof

Grade: C

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First one is routine chapter-1 for first-years, second one is easy using induction

**This proof has been marked as an page requiring an easy proof**## See also

- Distributivity of unions across intersections (almost the same: [ilmath]A\cup(B\cap C)=(A\cup B)\cap(A\cup C)[/ilmath])

## References