\ch{Sets, Proofs, and Functions}

Alright, we're done with Booleans! Sort of. The next thing we are going to look
at are \xti{sets}.

Sets were first studied by Georg Cantor, a German mathematician, in the second
half of the nineteenth century.  Back in his own day, the results Cantor found
by studying sets were considered so thoroughly bizarre that many of his
colleagues simply refused to believe that Cantor could be right.  In the end,
Cantor turned out to be right all along. His ideas can be found in any
introductory text on mathematics---including this one.

Sets are basically like lists--- think ``your grocery list'' or ``your
to-do-list'' --- except there's no multiplicity, and there's no intrinsic
order. A ``list'' is exactly what you think it is. It's a bunch of things. The
standard notation is to use $\mset{\mathrm{Braces}}$ for sets, and
$\mlist{\mathrm{Parentheses}}$ for lists. Lists can have duplication, and order
does matter.

$\mlist{4}$, $\mlist{4,4}$, and $\mlist{4,4,4}$ are all different \xtb{lists};
$\mset{4}$, $\mset{4,4}$, and $\mset{4,4,4}$ are the same \xtb{set}. In a
\xtb{list}, each $4$ is considered a separate item. In a \xtb{set}, $4$ can
appear a billion times, but it's only counted once.

$\mlist{1,2,3}$, $\mlist{3,1,2}$, and $\mlist{2,3,1}$ are all different
\xtb{lists}. $\mset{1,2,3}$, $\mset{3,1,2}$, and $\mset{2,3,1}$ are all the same
\xtb{set}. In a \xtb{set}, the order in which items appear doesn't matter; all
that matters is that the items are there. In a \xtb{list}, however, the order is
important.

In a list, order and multiplicity matter. In a set, order and multiplicity are
ignored. If you can't remember whether to use braces \{the curly things\}, or
parentheses (the round things), remember: \xti{a \xtb{brace} is used to
  \xtb{set} a broken bone.} I don't have a horrible pun having to do with
parentheses and lists, and for that, I apologize.

\input{3/1-elements-subsets.ltx}
\input{3/2-operators-functions.ltx}
\input{3/3-unions-intersections.ltx}
\input{3/4-natural-numbers.ltx}