Cartesian Product of Intersections

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

$\displaystyle \paren{ \prod_{i \mathop \in I} S_i } \cap \paren{ \prod_{i \mathop \in I} T_i } = \prod_{i \mathop \in I} \paren{S_i \cap T_i}$

Proof

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

$\blacksquare$