# Class of sets closed under complements properties

From Maths

## Contents

## Theorem statement

If [ilmath]\mathcal{A} [/ilmath] is a system of subsets of [ilmath]\Omega[/ilmath] such that^{[1]}:

- [math]\forall A\in\mathcal{A}[A^c\in\mathcal{A}][/math] where denotes the complement of [ilmath]A[/ilmath] - That is to say "[ilmath]\mathcal{A} [/ilmath] is closed under complements"

Then we have:^{[Note 1]}

- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cap[/ilmath]-closed [ilmath]\iff[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\cup[/ilmath]-closed
- [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cap[/ilmath]-closed [ilmath]\iff[/ilmath] [ilmath]\mathcal{A} [/ilmath] is [ilmath]\sigma[/ilmath]-[ilmath]\cup[/ilmath]-closed

## See also

## Notes

- ↑ See Index of properties under "closed" for the exact meanings of these

## References

- ↑
^{1.0}^{1.1}Probability Theory - A comprehensive course - Achim Klenke