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}
Children
Canonical set projection
Complex cartesian product
Cylinder set
Set of maps
Results
Bijection between parenthesis-sliced cartesian triple products
Cardinality of a finite set raised to a finite power
Cardinality of cartesian triple products is invariant under insertion of parentheses
Cardinality of finite cartesian products is invariant under insertion of parentheses
Cartesian product is not associative
Countable cartesian product with empty set is empty
Finite cartesian product with empty set is empty