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generalized Cartesian product (Definition)

Given any family of sets $ \{A_j\}_{j \in J}$ indexed by an index set $ J$, the generalized Cartesian product

$\displaystyle \prod_{j \in J} A_j $
is the set of all functions
$\displaystyle f\colon 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 projection map

$\displaystyle \pi_i\colon \prod_{j \in J} A_j \to A_i $
is the function defined by
$\displaystyle \pi_i(f) := f(i). $

The generalized Cartesian product is the product in the category of sets.

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.



"generalized Cartesian product" is owned by Mathprof. [ full author list (3) | owner history (1) ]
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See Also: Cartesian product, product topology, axiom of choice, ordered tuplet

Also defines:  projection map
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Cross-references: Cartesian product, axiom of choice, category of sets, functions, index set, indexed by
There are 38 references to this entry.

This is version 10 of generalized Cartesian product, born on 2001-10-19, modified 2006-10-21.
Object id is 361, canonical name is GeneralizedCartesianProduct.
Accessed 13284 times total.

Classification:
AMS MSC: 03E20 (Mathematical logic and foundations :: Set theory :: Other classical set theory )
Pending Errata and Addenda
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Cartesian product by pahio on 2004-09-30 13:40:05

Why the generalized Cartesian product is here denoted by using the \prod (= upper-case pi or russian "p") symbol? I think it does not tally with the cross symbol used in Cartesian product of two sets. I have seen that some textbooks and some teachers in university use the more logical \bigtimes symbol. Is the cause of the PM practice that the PM LaTeX does not yet contain the \bigtimes symbol?
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