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:
“O for a man who is a man, and, as my neighbor says, has a bone in his back which you cannot pass your hand through! Our statistics are at fault: the population has been returned too large. How many men are there to a square thousand miles in this country? Hardly one.”
—Henry David Thoreau (1817–1862)
“The older I get, the greater power I seem to have to help the world; I am like a snowball—the further I am rolled the more I gain.”
—Susan B. Anthony (1820–1906)