Section 8.3 Cartesian products of sets
Now when we were defining sets we used lists
\(A = \set{1,2,3}\) and the order in which we put things in the list didn’t matter. On some occasions we really need to put things into some order; we need a way to write an ordered pair of elements in which one of the pair is the
first and the other is the
second. Coordinates of points,
\((x,y)\text{,}\) in the plane are a very good example of this: the first number is the
\(x\)-coordinate (horizontal position) and the second is the
\(y\)-coordinate (vertical position), and we should not mix them up
99 . The point
\((1,13)\) on the plane is not the same as the point
\((13,1)\text{.}\) We are used to this notation, “
\((x,y)\)”, but we should define it before we go any further.
Definition 8.3.1.
An ordered pair of elements is an ordered list of two elements. We write this as \((a,b)\) with round brackets rather than braces. Ordered pairs have the properties that
\((a,b) \neq (b,a)\) unless \(a=b\text{,}\) and
\((a,b) = (c,d)\) only when \((a=c)\) and \((b=d)\text{.}\)
Given two sets \(A,B\text{,}\) the set of all possible ordered pairs is the Cartesian product of those sets. To be more precise:
Definition 8.3.2. Cartesian product.
Let \(A,B\) be sets. The Cartesian product,or just product, of \(A\) and \(B\) is the set of all ordered pairs \((a,b)\) such that \(a \in A\) and \(b \in B\text{.}\) We write this as \(A \times B\text{.}\)
\begin{gather*}
A \times B = \set{ (a,b) : a\in A \text{ and } b \in B }
\end{gather*}
Note that \(A \times B \neq B \times A\text{,}\) unless \(A=B\text{.}\)
Example 8.3.3.
If we set \(A = \set{a,b,c}\) and \(B=\set{1,2}\) then
\begin{align*}
A\times B &= \set{ (a,1), (a,2), (b,1), (b,2), (c,1), (c,2)}
\end{align*}
So we are used to playing with cartesian products in the context of functions —
\(\mathbb{R} \times \mathbb{R}\) is the whole
cartesian plane 100 and functions we are used to are just subsets of this. For example, the parabola
\(y=x^2\) is the set
\begin{gather*}
\set{ (x,y) \in \mathbb{R} \times \mathbb{R} \so y=x^2}
\end{gather*}
which is a subset of \(\mathbb{R} \times \mathbb{R}\text{.}\)
Mind you, people rarely call the parts of an \(x,y\)-coordinate by their correct names. The \(x\) (ie the first of the pair) is called the abscissa and its use goes back at least as far as Fibonacci. The \(y\) (the second of the pair) is the ordinate. These terms are not so common in modern English and people typically just call them \(x\)-coordinate and \(y\)-coordinate (which is a little jarring to the ear of the pedant).
While this is named for the French mathematician and philosopher Rene Descartes (1596 – 1650), it was also invented by Pierre de Fermat (1601 – 1665), and even earlier by Nicole Oresme (1325 – 1382). Fermat is famous for writing his “Last Theorem” in the margin of a book, and Oresme was the first to prove that the infinite series \(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots \) does not converge.
