Duality Principle for Sets

Theorem

Any identity in set theory which uses any or all of the operations:

• intersection $\cap$
• union $\cup$
• Empty set $\varnothing$
• Universal set $\mathbb U$

and none other, remains valid if:

• $\cap$ and $\cup$ are exchanged throughout
• $\varnothing$ and $\mathbb U$ are exchanged throughout.

Proof

Follows from:

• Algebra of Sets is a Boolean Ring
• Principle of Duality of Boolean Rings

$\blacksquare$

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 5$: Complements and Powers
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.5$