Duality Principle for Sets

This proof is about Duality Principle in the context of Set Theory. For other uses, see Duality Principle.

Theorem

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

Set intersection $\cap$
Set union $\cup$
Empty set $\O$
Universal set $\mathbb U$

and none other, remains valid if:

$\cap$ and $\cup$ are exchanged throughout
$\O$ and $\mathbb U$ are exchanged throughout.


Proof

Follows from:

  • Algebra of Sets is Huntington Algebra
  • Principle of Duality of Huntington Algebras

$\blacksquare$


Sources

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