The Cartesian Product

We have encountered the definition of an ordered pair ${(a,b)}$ when studying the pairing axiom. We could form a set of all order pairs by defining a cartesian product between two sets, or even of a set and itself.

Definition: The cartesian product of two sets ${A}$ and ${B}$ is the set of all ordered pairs where ${a}$ is an element of ${A}$ and ${b}$ is an element of ${B}$ .

$\displaystyle A \times B = \forall a \forall b\{(a,b):a\in A \land b \in B\} \ \ \ \ \ (1)$

But is this a set? To answer this we need the concept of the Power Set of a set ${X}$ .

Definition: The Power Set ${\wp(X)}$ is the set of all subsets of ${X}$ .

$\displaystyle \forall X \exists Y \forall u (u\in Y \iff u \subset X) \ \ \ \ \ (2)$

So we can say ${\wp(x)=\{u : u\subset X\}}$

If we have for all ${x}$ in ${X}$ and for all ${y}$ in ${Y}$ the set ${Z = X\cup Y}$ the we know that ${\{x\}}$ is an element of the Power Set of ${Z}$ by its definition. Similarly, we also know that ${\{x,y\}}$ is an element of the Power Set of ${Z}$ . Hence noting that ${\{\{x\},\{x,y\}\}=(x,y)}$ :

$\begin{equation*} \{\{x\},\{x,y\}\} \in \wp \wp (X) \implies (x,y) \in \wp \wp (Z) \end{equation*}$

If ${(x,y)}$ is in the power set of the power set of ${Z}$ then the set of all ${(x,y)=X \times Y}$ is a subset of the power set of the power set of ${Z}$ . From above ${Z=X\cup Y}$ .

$\begin{equation*} X \times Y \subset \wp \wp (X \cup Y) \end{equation*}$

Hence we know the Power Set is a set and we know that a subset of a set is a set and so the cartesian product forms a set. All is good!

The cartesian product has many uses. At first encountered we may know it as a graph ${(x,y)}$ that can be plotted i.e. a set of coordinates. This idea leads to an isomorphic map from cartesian products of sets of real numbers to vectors under a given basis. Ordered pairs can be used to describe relations out of which come mappings and functions. The set theory used in the above is profound and powerful in nature.