Verse and Dimensions Wikia
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Verse and Dimensions Wikia

The Cartesian product of two sets A and B is the set of all possible ordered pairs (a, b) such that a is an element of A and b is an element of B (using set-builder notation, Cartesian products can be written as ). The cardinality of the Cartesian product of two sets is always the product of the cardinalities of the two sets.

The Cartesian product of two shapes A and B, is a shape whose coordinates are (a, b), where a is a coordinate of shape A and b is a coordinate of shape B. The dimensionality of the resulting shape is the sum of the dimensionalities of A and B.

Cartesian products of multiple sets are known as "n-fold Cartesian products" or "n-ary Cartesian products", and will result in a set of n-tuples. Cartesian powers are a unary operation defined by repeated cartesian products. The Cartesian power of a set is written , where A is a set and n is some non-negative integer. means , and there is a bijection between the cardinality of and the cardinality of the set of functions from a set with n elements and A. If A has a cardinality of 2, then the cardinality of is equal to the cardinality of the power set of a set with n elements.

Repeated cartesian products of n Euclidean lines will result in n-dimensional Euclidean spaces.

The Cartesian product of a shape (typically a polytope) and a line segment is the shape's prism.

Repeated cartesian products of a line segment will create a hypercube.

Repeated cartesian products of hyperballs will create a rotatope. If hyperspheres are to be included in the Cartesian products, then toratopes will be the result.

Taking the Cartesian product of two shapes (typically polytopes) results in a duoprism. The Cartesian product of some shape and a point is the original shape.

The Cartesian product of any set (which can be a shape) and the empty set is the empty set. This is because the cardinality of the empty set is 0, and any number (in this case, the cardinality of the set/shape) multiplied by 0 is 0, including the cardinality of the continuum.

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