Class Intersection Distributes over Class Union
Jump to navigation
Jump to search
Theorem
Let $A$, $B$ and $C$ be classes.
Then:
- $A \cap \paren {B \cup C} = \paren {A \cap B} \cup \paren {A \cap C}$
where:
- $A \cap B$ denotes class intersection
- $B \cup C$ denotes class union.
Proof
\(\ds \) | \(\) | \(\ds x \in A \cap \paren {B \cup C}\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in A \land \paren {x \in B \lor x \in C}\) | Definition of Class Union and Definition of Class Intersection | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \paren {x \in A \land x \in B} \lor \paren {x \in A \land x \in C}\) | Conjunction is Left Distributive over Disjunction | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds x \in \paren {A \cap B} \cup \paren {A \cap C}\) | Definition of Class Union and Definition of Class Intersection |
$\blacksquare$
Also see
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 5$ The union axiom: Exercise $5.6. \ \text {(a)}$