159 Cartesian Product

A set of all ordered pairs formed from two sets that is used to construct product domains and coordinate sets.

definition (Cartesian Product) A set consisting of all possible ordered combinations of elements from a collection of sets, denoted by \(X \times Y\) for two sets and by \(X_1 \times \dots \times X_n\) for \(n\) sets, where the following conditions apply:

For two sets, the product contains all ordered pairs \((x, y)\) where
the first component is from the first set and the second component is from the second.
For a finite collection, the product consists of all ordered \(n\)-tuples \((x_1, \dots, x_n)\) where
each \(x_i\) is an element of the corresponding factor \(X_i\).
For an arbitrary indexed family, the product is the set of all functions \(x\) from the index set \(A\) to the union of the sets such that \(x(\alpha) \in X_\alpha\) for each \(\alpha \in A\).

where

\(X, Y, X_i, X_\alpha\) are non-empty sets
\((x, y)\) is an ordered pair
\((x_1, \dots, x_n)\) is an ordered \(n\)-tuple
\(A\) is the index set used to label the family of sets

Note:

\(A\) is also written \(\Lambda\).
for an indexed family the product is also written \(\prod_{\alpha \in A} X_\alpha\).

159.1 Elementary Example

159.1.1 Simple

The Cartesian product \(X \times Y\) is the set of all ordered pairs with first entry from \(X\) and second entry from \(Y\).

\[ X = \{ 1,\ 2,\ 3 \} \]

\[ Y = \{ a,\ b \} \]

\[ X \times Y = \{ (1,a),\ (1,b),\ (2,a),\ (2,b),\ (3,a),\ (3,b) \} \]

where

\(X\) and \(Y\) are the factor sets.
\((x,y)\) is an ordered pair.

159.1.2 General

For three sets the product is the set of ordered triples.

\[ X = \{ 1,\ 2 \},\quad Y = \{ a,\ b \},\quad Z = \{ p,\ q \} \]

\[ X \times Y \times Z = \{ (1,a,p),\ (1,a,q),\ (1,b,p),\ (1,b,q),\ (2,a,p),\ (2,a,q),\ (2,b,p),\ (2,b,q) \} \]

where

\(X \times Y \times Z\) is the three-fold Cartesian product.

158 Bundle

160 Constant Metric Tensor