Cartesian Product of Intersections
Jump to navigation
Jump to search
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
- $\ds \paren {\prod_{i \mathop \in I} S_i} \cap \paren {\prod_{i \mathop \in I} T_i} = \prod_{i \mathop \in I} \paren {S_i \cap T_i}$
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 |
$\blacksquare$
Sources
- 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 $1.2 \ \text{(n)}$