| Version current |
Version 9 |
| Given any family of sets $\{A_j\}_{j \in J}$ indexed by an index set $J$, the \emph{generalized Cartesian product} |
Given any family of sets $\{A_j\}_{j \in J}$ indexed by an index set $J$, the \emph{generalized Cartesian product} |
| \[ |
\[ |
| \prod_{j \in J} A_j |
\prod_{j \in J} A_j |
| \] |
\] |
| is the set of all functions |
is the set of all functions |
| \[ |
\[ |
| f\colon J \to \bigcup_{j \in J} A_j |
f\colon J \to \bigcup_{j \in J} A_j |
| \] |
\] |
| such that $f(j) \in A_j$ for all $j \in J$. |
such that $f(j) \in A_j$ for all $j \in J$. |
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| For each $i \in J$, the \emph{projection map} |
For each $i \in J$, the \emph{projection map} |
| \[ |
\[ |
| \pi_i\colon \prod_{j \in J} A_j \to A_i |
\pi_i\colon \prod_{j \in J} A_j \to A_i |
| \] |
\] |
| is the function defined by |
is the function defined by |
| \[ |
\[ |
| \pi_i(f) := f(i). |
\pi_i(f) := f(i). |
| \] |
\] |
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| The generalized Cartesian product is the \PMlinkname{product}{CategoricalDirectProduct} in the category of sets. |
The generalized Cartesian product is the \PMlinkname{product}{CategoricalDirectProduct} in the category of sets. |
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| The axiom of choice is the statement that the generalized Cartesian product of nonempty sets is nonempty. |
The axiom of choice is the statement that the generalized Cartesian product of nonempty sets is nonempty. |
| The generalized Cartesian product is usually called the Cartesian product. |
The generalized Cartesian product is usually called the Cartesian product. |
