Union Distributes over Intersection/Proof 1

Theorem

Set union is distributive over set intersection:

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

Proof

\(\ds \)

\(\)

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

\(\ds \)

\(\leadstoandfrom\)

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

Definition of Set Union and Definition of Set Intersection

\(\ds \)

\(\leadstoandfrom\)

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

Disjunction is Left Distributive over Conjunction

\(\ds \)

\(\leadstoandfrom\)

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

Definition of Set Union and Definition of Set Intersection

$\blacksquare$

Sources

• 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 $3.1$