Class of sets closed under complements properties

Theorem statement

If $\mathcal{A}$ is a system of subsets of $\Omega$ such that:

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

Then we have:

• $\mathcal{A}$ is $\cap$-closed $\iff$ $\mathcal{A}$ is $\cup$-closed
• $\mathcal{A}$ is $\sigma$-$\cap$-closed $\iff$ $\mathcal{A}$ is $\sigma$-$\cup$-closed

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Proof:

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TODO: see page 2 if help needed - basically is just De Morgan's laws

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See also

• Types of set algebras
• Class of sets closed under set-subtraction properties