# Cartesian Product

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Projection of a Torus

The Cartesian Product is a particular way of combining two sets. If we have two sets A and B, the Cartesian product is the set of all ordered pairs created by listing an arbitrary element of A first, then an element of B. Note that order matters here, and the Cartesian Product $A \times B$ is different from $B \times A$. In the former, the elements of A are listed first, and in the latter, the elements of B are listed first.

An example is $\{1,4,6\} \times \{2,9\} = \{ (1,2), (1,9), (4,2), (4, 9), (6,2), (6,9)\}$

Note that this is different from $\{2,9\} \times \{1,4,6\} = \{ (2,1), (2,4), (2,6), (9,1), (9,4), (9,6)\}$.

Another example is $\{a,b,c\} \times \{2, 5\} = \{(a,2), (a, 5), (b,2), (b,5), (c,2), (c, 5)\}$

A more abstract example is the complex numbers can be thought of as the Cartesian product of the real numbers with the imaginary numbers. The first component of a complex number is its real part, and the second component is its imaginary part.