Cartesian Product of Intersections/Corollary 2
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Corollary to Cartesian Product of Intersections
- $\paren {A \times B} \cap \paren {B \times A} = \paren {A \cap B} \times \paren {A \cap B}$
Proof
Take the result Cartesian Product of Intersections:
- $\paren {S_1 \cap S_2} \times \paren {T_1 \cap T_2} = \paren {S_1 \times T_1} \cap \paren {S_2 \times T_2}$
Put $S_1 = A, S_2 = B, T_1 = B, T_2 = A$:
\(\ds \) | \(\) | \(\ds \paren {A \times B} \cap \paren {B \times A}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {A \cap B} \times \paren {B \cap A}\) | Cartesian Product of Intersections | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {A \cap B} \times \paren {A \cap B}\) | Intersection is Commutative |
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $1$. Sets: Exercise $10$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 3$. Ordered pairs; cartesian product sets: Exercise $3 \ (3)$