Chapter 12 Cardinality
This last chapter of the text brings together many of the ideas and techniques that we have learned to make sense of cardinality — the size of a set. When a set is finite the cardinality is a very intuitive concept — just 162 count up the elements:
\begin{equation*}
|\set{1,3,7,18,53}| = 5.
\end{equation*}
By carefully describing how we count elements in finite sets in terms of bijections we can extend our understanding of cardinality to infinite sets. This allows us to make sense of statements such as
\begin{equation*}
|\mathbb{Q}| = |\mathbb{Z}|
\end{equation*}
which are, to say the least, quite counter-intuitive. This also enables us to prove Cantor’s Theorem 12.4.3. This result is arguably one of the most important pieces of mathematics that can be proven in undergraduate mathematics. It tells us something fundamental about the nature of infinity: not only are there different sorts of infinities, but there are an infinite number of different infinities!
The reader is right to be a little skeptical of the use of the word “just” here; counting up the elements of a finite set can sometimes be quite difficult. Here we have listed out all the elements quite explicitly, but somes a set will be defined more implicitly, and then counting the elements can be quite challenging. The interested reader should search-engine their way to a description of enumerative combinatorics which is the mathematics of counting.
