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'generalized cartesian product'
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| Title of object: |
generalized cartesian product |
| Canonical Name: |
GeneralizedCartesianProduct |
| Type: |
Definition |
| Created on: |
2001-10-19 00:54:46 |
| Modified on: |
2003-05-31 13:02:52 |
| Classification: |
msc:03E20 |
| Defines: |
projection map |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic} |
Content:
Given any family of sets $\{A_j\}_{j \in J}$ indexed by an index set $J$, the {\em generalized cartesian product}
\prod_{j \in J} A_j
is the set of all functions
f: J \to \bigcup_{j \in J} A_j
such that $f(j) \in A_j$ for all $j \in J$.
For each $i \in J$, the {\em projection map} $$\pi_i: \prod_{j \in J} A_j \to A_i$$ is the function defined by
\pi_i(f) := f(i).
$$ |
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