Cartesian Product Distributes over Union
(Redirected from Cartesian Product of Unions/Corollary)
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Theorem
Cartesian product is distributive over union:
- $A \times \paren {B \cup C} = \paren {A \times B} \cup \paren {A \times C}$
- $\paren {B \cup C} \times A = \paren {B \times A} \cup \paren {C \times A}$
Proof
Take the result Cartesian Product of Unions:
- $\paren {S_1 \cup S_2} \times \paren {T_1 \cup T_2} = \paren {S_1 \times T_1} \cup \paren {S_2 \times T_2} \cup \paren {S_1 \times T_2} \cup \paren {S_2 \times T_1}$
Put $S_1 = S_2 = A, T_1 = B, T_2 = C$:
\(\ds \) | \(\) | \(\ds A \times \paren {B \cup C}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {A \cup A} \times \paren {B \cup C}\) | Set Union is Idempotent | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {A \times B} \cup \paren {A \times C} \cup \paren {A \times C} \cup \paren {A \times B}\) | Cartesian Product of Unions | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {A \times B} \cup \paren {A \times C}\) | Set Union is Idempotent |
Thus:
- $A \times \paren {B \cup C} = \paren {A \times B} \cup \paren {A \times C}$
The other result is proved similarly.
$\blacksquare$
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 6$: Ordered Pairs: Exercise $\text{(i)}$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $1$. Sets: Exercise $8 \ \text{(ii)}$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 3$. Ordered pairs; cartesian product sets: Exercise $3 \ (2)$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $15$