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 $18$