Symmetric edits

Advanced/Symmetric Group/parts/0 intro.tex

@@ -1,18 +1,7 @@
 \section{Introduction}
 
-\definition{Intuitive permutations}
-Intuitively, a \textit{permutation} is an ordered arrangement of a set of objects. \par
-For example, $123$, $312$, and $231$ are all permutations of 1, 2, and 3.
 
-\problem{}
-List all permutations on three objects. \par
-How many permutations of $n$ objects are there?
-
-\vfill
-
-
-
-\definition{Formal permutations}<permadef>
+\definition{}
 Let $\Omega$ be an arbitrary set of $n$ objects. \par
 A \textit{permutation} on $\Omega$ is a bijective map $f: \Omega \to \Omega$.
 
@@ -26,24 +15,31 @@ The permutation $[312]$ is given by a map $f$ defined by the following table:
 	\item $f(3) = 2$
 \end{itemize}
 
-Similarly, the \textit{trivial permutation} $[123]$ is given by the identity map $f(x) = x$.
 
 \problem{}
-What map corresponds to the permutation $[321]$?
+List all permutations on three objects. \par
+How many permutations of $n$ objects are there?
 
 \vfill
 
 
+\problem{}
+What map corresponds to the permutation $[321]$?
+
+
+\vfill
 
 \problem{}
-Why do we define permutations as a \textit{bijective} map?
+What map corresponds to the \say{do-nothing} permutation? \par
+Write it as a function and in square-bracket notation. \par
+\note[Note]{We usually call this the \textit{trivial permutation}}
 
 
 \vfill
 \pagebreak
 
 
-We can visualize permutations with a diagram we'll call the \say{braid.}
+We can visualize permutations with a \textit{string diagram}, shown below. \par
 The arrows in this diagram denote the image of $f$ for each possible input.
 Two examples are below:
 
@@ -161,8 +157,8 @@ The rightmost diagram uses arbitrary, meaningless labels.
 It shouldn't be hard to see that despite the different \say{output} order (2134 and 1432), \par
 the same permutation is depicted in all three diagrams. This example demonstrates two things:
 \begin{itemize}[itemsep=2mm]
-	\item First, the items of our set do not have any meaning. \par
-	$\Omega$ is just a set of arbitrary \textit{things}, which we may label however we like.
+	\item First, the names of the items in our set do not have any meaning. \par
+	$\Omega$ is just a set of $n$ arbitrary things, which we may label however we like.
 
 	\item Second, permutations are verbs. We do not care about the \say{output} of a certain permutation,
 	we care about what it \textit{does}. We could, for example, describe the permutation above as
@@ -176,16 +172,10 @@ Why, then, do we order our elements when we talk about permutations? As noted be
 If we assign a natural order to the elements of $\Omega$ (say, 1234), we can identify permutations by simply listing
 their output:
 Clearly, $[1234]$ represents the trivial permutation, $[2134]$ represents \say{swap first two,}
-and $[4123]$ represents \say{cycle left.}
+and $[4123]$ represents \say{cycle right.}
 
 \problem{}
-Draw braids for $[4123]$ and $[2341]$.
+Draw string diagrams for $[4123]$ and $[2341]$.
 
 \vfill
-
-
-Finally, note that permutations (as defined in \ref{permadef}) are \textit{not} \say{orderings of a certain set.} \par
-They are defined as \textit{bijective maps}, which can be written as orderings of a given array. \par
-Remember: permutations are verbs!
-
 \pagebreak