Cartesian product

The Cartesian product of sets is the set of ordered tuples formed by choosing one element from each set. The order of the sets determines the positions in each tuple.

Set: A collection of distinct objects treated as a mathematical object. A Cartesian product is constructed from sets as its input collections.
Projection (mathematics): A function that extracts a specified coordinate from an ordered tuple. Each coordinate projection maps a product back to one of its factor sets.
Relation (mathematics): A relation from one set to another is a subset of their Cartesian product. The product supplies all possible ordered pairs from which a relation selects.
Direct product: A product construction combining algebraic structures coordinate by coordinate. It specializes product structure to objects such as groups, rings, and modules.
Ordered pair: A pair of objects in which the first and second positions are distinguished. The product of two sets consists of ordered pairs, not unordered combinations.
Finite set: A set containing a fixed, nonnegative number of elements. For finite factors, the product size is the product of their cardinalities.
Coordinate space: A space whose points are represented by ordered lists of coordinates. Its points are tuples drawn from products of coordinate sets.
Disjoint union: A union construction that keeps elements from different sets distinguishable by their source. Unlike a product, it collects alternatives from factors rather than tuples choosing from all factors.
Tuple: A finite sequence of objects whose positions are distinguished. Products of three or more sets contain tuples with one position for each set.
Empty set: The set containing no elements. A product with an empty factor is empty because no tuple can choose from it.

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