Cartesian Product of Intersections

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Theorem

$\paren {S_1 \cap S_2} \times \paren {T_1 \cap T_2} = \paren {S_1 \times T_1} \cap \paren {S_2 \times T_2}$

where $S_1, S_2, T_1, T_2$ are sets.


Corollary 1

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


Corollary 2

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


General Case

$\displaystyle \left({ \prod_{i \mathop = 1}^n S_i }\right) \cap \left({ \prod_{i \mathop = 1}^n T_i }\right) = \prod_{i \mathop = 1}^n \left({S_i \cap T_i}\right)$


Proof

\(\displaystyle \) \(\) \(\displaystyle \tuple {x, y} \in \paren {S_1 \cap S_2} \times \paren {T_1 \cap T_2}\)
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \paren {x \in S_1 \land x \in S_2} \land \paren {y \in T_1 \land y \in T_2}\) Definition of Cartesian Product and Definition of Set Intersection
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \paren {x \in S_1 \land y \in T_1} \land \paren {x \in S_2 \land y \in T_2}\) Rule of Commutation, Rule of Association
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \tuple {x, y} \in S_1 \times T_1 \land \tuple {x, y} \in S_2 \times T_2\) Definition of Cartesian Product
\(\displaystyle \) \(\leadstoandfrom\) \(\displaystyle \tuple {x, y} \in \paren {S_1 \times T_1} \cap \paren {S_2 \times T_2}\) Definition of Set Intersection

$\blacksquare$


Sources