# Duality Principle for Sets

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*This proof is about the Duality Principle for Sets. 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 $\varnothing$
- 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:

$\blacksquare$

## 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