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:
“A man who is good enough to shed his blood for his country is good enough to be given a square deal afterwards. More than that no man is entitled to, and less than that no man shall have.”
—Theodore Roosevelt (1858–1919)
“He will deliver you from six troubles; in seven no harm shall touch you. In famine he will redeem you from death, and in war from the power of the sword. You shall be hidden from the scourge of the tongue, and shall not fear destruction when it comes. At destruction and famine you shall laugh, and shall not fear the wild animals of the earth. For you shall be in league with the stones of the field, and the wild animals shall be at peace with you.”
—Bible: Hebrew, Job 5:19-23.