Cartesian Product of Intersections/Corollary 2

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 $3 \ (3)$