ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
▼ Theory
▼ Zermelo-Fraenkel set theory
▼ Set
▼ Binary cartesian set product
▼ Binary relation
▼ Map
Definition D326
Cartesian product
Formulation 0
Let $X_j$ be a D11: Set for each $j \in J$ such that
(i) $\bigcup_{j \in J} X_j$ is the D77: Set union of $X = \{ X_j \}_{j \in J}$
The cartesian product of $X = \{ X_j \}_{j \in J}$ is the D11: Set \begin{equation} \prod_{j \in J} X_j : = \left\{ x : J \to \bigcup_{j \in J} X_j \mid \forall \, j \in J : x_j \in X_j \right\} \end{equation}
Also known as
Set direct product
Children
▶ D327: Canonical set projection
▶ D5639: Complex cartesian product
▶ D2758: Cylinder set
▶ D68: Set of maps
Results
▶ R4630: Bijection between parenthesis-sliced cartesian triple products
▶ R4315: Cardinality of a finite set raised to a finite power
▶ R4632: Cardinality of cartesian triple products is invariant under insertion of parentheses
▶ R4631: Cardinality of finite cartesian products is invariant under insertion of parentheses
▶ R221: Cartesian product is not associative
▶ R4263: Countable cartesian product with empty set is empty
▶ R4262: Finite cartesian product with empty set is empty