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)
“... whilst you are proclaiming peace and good will to men, Emancipating all Nations, you insist upon retaining absolute power over wives. But you must remember that Arbitrary power is like most other things which are very hard, very liable to be broken—and notwithstanding all your wise Laws and Maxims we have it in our power not only to free ourselves but to subdue our Masters, and without violence throw both your natural and legal authority at our feet ...”
—Abigail Adams (1744–1818)