Union Distributes over Intersection/Proof 1

Theorem

Set union is distributive over set intersection:

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

$\ds $

$\ds x \in R \cup \paren {S \cap T}$

$\ds $

$\leadstoandfrom$

$\ds x \in R \lor \paren {x \in S \land x \in T}$

Definition of Set Union and Definition of Set Intersection

$\ds $

$\leadstoandfrom$

$\ds \paren {x \in R \lor x \in S} \land \paren {x \in R \lor x \in T}$

$\ds $

$\leadstoandfrom$

$\ds x \in \paren {R \cup S} \cap \paren {R \cup T}$

Definition of Set Union and Definition of Set Intersection

$\blacksquare$

1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 4$: Unions and Intersections
1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 3$: Unions and Intersections of Sets: Theorem $3.1$