# Union Distributes over Intersection/Proof 2

## Theorem

Set union is distributive over set intersection:

- $R \cup \paren {S \cap T} = \paren {R \cup S} \cap \paren {R \cup T}$

## Proof

From Intersection Distributes over Union:

- $R \cap \paren {S \cup T} = \paren {R \cap S} \cup \paren {R \cap T}$

From the Duality Principle for Sets, exchanging $\cup$ for $\cap$ throughout, and vice versa, reveals the result:

- $R \cup \paren {S \cap T} = \paren {R \cup S} \cap \paren {R \cup T}$

$\blacksquare$

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.5$. The algebra of sets: Example $18$

- 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Operations - 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 2$ - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 2$ - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): Appendix $\text{A}.2$: Theorem $\text{A}.10$