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.

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

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

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

$\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 $

$\leadstoandfrom$

$\ds \paren {x \in S_1 \land y \in T_1} \land \paren {x \in S_2 \land y \in T_2}$

$\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

$\blacksquare$

1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 6$: Ordered Pairs: Exercise $\text{(ii)}$
1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 9 \alpha$
1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.5$: Products: Definition $3.5.1$
1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $14$
2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts: Exercise $1.2 \ \text{(n)}$