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:
“Rationalists, wearing square hats,
Think, in square rooms,
Looking at the floor,
Looking at the ceiling.
They confine themselves
To right-angled triangles.”
—Wallace Stevens (1879–1955)
“‘Tis not such lines as almost crack the stage
When Bajazet begins to rage;
Nor a tall met’phor in the bombast way,
Nor the dry chips of short-lunged Seneca.
Nor upon all things to obtrude
And force some odd similitude.
What is it then, which like the power divine
We only can by negatives define?”
—Abraham Cowley (1618–1667)