Cartesian Product of Intersections/Corollary 1

Corollary to Cartesian Product of Intersections

$A \times \paren {B \cap C} = \paren {A \times B} \cap \paren {A \times C}$

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 $S_1 = S_2 = A, T_1 = B, T_2 = C$:

\(\ds A \times \paren {B \cap C}\)

\(=\)

\(\ds \paren {A \cap A} \times \paren {B \cap C}\)

Set Intersection is Idempotent

\(\ds \)

\(=\)

\(\ds \paren {A \times B} \cap \paren {A \times C}\)

Cartesian Product of Intersections

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $1$. Sets: Exercise $8 \ \text{(i)}$
• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.9$: Cartesian Product: Theorem $9.2$
• 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 3$. Ordered pairs; cartesian product sets: Exercise $3 \ (1)$