Cartesian Square and Cartesian Power
The Cartesian square (or binary Cartesian product) of a set X is the Cartesian product X2 = X × X. An example is the 2-dimensional plane R2 = R × R where R is the set of real numbers - all points (x,y) where x and y are real numbers (see the Cartesian coordinate system).
The cartesian power of a set X can be defined as:
An example of this is R3 = R × R × R, with R again the set of real numbers, and more generally Rn.
The n-ary cartesian power of a set X is isomorphic to the space of functions from an n-element set to X. As a special case, the 0-ary cartesian power of X may be taken to be a singleton set, corresponding to the empty function with codomain X.
Read more about this topic: Cartesian Product
Famous quotes containing the words square and/or power:
“This house was designed and constructed with the freedom of stroke of a forester’s axe, without other compass and square than Nature uses.”
—Henry David Thoreau (1817–1862)
“What preoccupies us, then, is not God as a fact of nature, but as a fabrication useful for a God-fearing society. God himself becomes not a power but an image.”
—Daniel J. Boorstin (b. 1914)