more typo fixes

Author: oscarlevin

source/ch_graphtheory.ptx

@@ -4,11 +4,11 @@
   <title>Graph Theory</title>
   <introduction>
     <p>
-      Graph theory has existed as a branch of mathematics for only a short time; the first book on graph theory was published less than 100 years ago.  While the first problem related to what we now call graph theory dates back to 1735, it has been the advent of computers that has shown that the subject's true utility.  It is a subject with simple beauty and surprising depth.  Many of the main areas of graph theory can be understood with almost no mathematical prerequisites, yet new research in the subject generate hundreds of peer-reviewed research papers each year.
+      Graph theory has existed as a branch of mathematics for only a short time; the first book on graph theory was published less than 100 years ago.  While the first problem related to what we now call graph theory dates back to 1735, it has been the advent of computers that has shown the subject's true utility.  It is a subject with simple beauty and surprising depth.  Many of the main areas of graph theory can be understood with almost no mathematical prerequisites, yet new research in the subject generates hundreds of peer-reviewed research papers each year.
     </p>
 
     <p>
-      In this chapter we will explore just a few of the ways you can use graphs and their properties to solve problems that show up in computer science, mathematics, and almost every other applied science.
+      In this chapter, we will explore just a few of the ways you can use graphs and their properties to solve problems that show up in computer science, mathematics, and almost every other applied science.
     </p>
   </introduction>
 

source/exercises/gt-trees.ptx

@@ -487,7 +487,7 @@
   <exercise>
     <statement>
       <p>
-        Prove that every connected graph which is not itself a tree must have at last three
+        Prove that every connected graph which is not itself a tree must have at least three
         different
         (although possibly isomorphic)
         spanning trees.

source/sec_gt-intro.ptx

@@ -94,15 +94,15 @@
       <p>
         Graph Theory is a relatively new area of mathematics,
         first studied by the super famous mathematician Leonhard Euler in 1735.
-        Since then it has blossomed in to a powerful tool used in nearly every branch of science and is currently an active area of mathematics research.
+        Since then it has blossomed into a powerful tool used in nearly every branch of science and is currently an active area of mathematics research.
       </p>
 
       <p>
         The problem above, known as the
         <em>Seven Bridges of Königsberg</em>,
         is the problem that originally inspired graph theory.
         Consider a <q>different</q> problem: Below is a drawing of four dots connected by some lines.
-        Is it possible to trace over each line once and only once (without lifting up your pencil,
+        Is it possible to trace over each line once and only once (without lifting your pencil,
         starting and ending on a dot)?
       </p>
 
@@ -139,19 +139,19 @@
         we say they are <term>adjacent</term>.
             <idx><h>adjacent</h><h>vertices</h></idx>
         The nice thing about looking at graphs instead of pictures of rivers,
-        islands and bridges is that we now have a mathematical object to study.
+        islands, and bridges is that we now have a mathematical object to study.
         We have distilled the <q>important</q>
-        parts of the bridge picture for the purposes of the problem.
+        parts of the bridge picture for the problem.
         It does not matter how big the islands are,
         what the bridges are made out of,
         if the river contains alligators, etc.
         All that matters is which land masses are connected to which other land masses,
         and how many times.
-        This was the great insight that Euler had.
+        This was the great insight Euler had.
       </p>
 
       <p>
-        We will return to the question of finding paths through graphs in <xref ref="sec_gt-paths"/>.  In this section we will explore various ways that graphs can be used to represent, or <em>model</em>, real world problems.  Along the way we will introduce some basic definitions, terminology, and notation that will be used in the rest of the chapter.
+        We will return to the question of finding paths through graphs in <xref ref="sec_gt-paths"/>.  In this section, we will explore various ways that graphs can be used to represent, or <em>model</em>, real-world problems.  Along the way, we will introduce some basic definitions, terminology, and notation that will be used in the rest of the chapter.
 
       </p>
       <worksheet xml:id="PA-gt-intro">
@@ -274,7 +274,7 @@
 
           <statement>
             <p>
-              First, let's just count the number of vertices and edges in each graph.
+              First, let's count the number of vertices and edges in each graph.
             </p>
             <p component="interactive">
               Number of vertices: <m>G_1</m>: <var name="10" width="5"/>; <m>G_2</m>: <var name="6" width="5"/>; <m>G_3</m>: <var name="6" width="5"/>; <m>G_4</m>: <var name="6" width="5"/>.
@@ -448,13 +448,13 @@
   <p>
     The way we avoid ambiguities in mathematics is to provide concrete and rigorous <em>definitions</em>.
     Crafting good definitions is not easy, but it is incredibly important. The definition is the
-    agreed upon starting point from which all truths in mathematics proceed. Is there a graph with
+    agreed-upon starting point from which all truths in mathematics proceed. Is there a graph with
     no edges? We have to look at the definition to see if this is possible. </p>
 
   <p> 
     We want our
     definition to be precise and unambiguous, but it also must agree with our intuition for the
-    objects we are studying. It needs to be useful: we <em>could</em> define a graph to be a six legged
+    objects we are studying. It needs to be useful: we <em>could</em> define a graph to be a six-legged
     mammal, but that would not let us solve any problems about bridges. Instead, here is the (now)
     standard definition of a graph. 
   </p>
@@ -573,8 +573,7 @@
     <solution>
       <p> No. Here the vertex sets of each graph are equal, which is a
         good start. Also, both graphs have two edges. In the first graph, we have edges <m>\{a,b\}</m>
-        and <m>\{b,c\}</m>, while in the second graph we have edges <m>\{a,c\}</m> and <m>\{c,b\}</m>. Now we
-        do have <m>\{b,c\} = \{c,b\}</m>, so that is not the problem. The issue is that <m>\{a,b\} \ne
+        and <m>\{b,c\}</m>, while in the second graph we have edges <m>\{a,c\}</m> and <m>\{c,b\}</m>. Of course, <m>\{b,c\} = \{c,b\}</m>, so that is not the problem. The issue is that <m>\{a,b\} \ne
         \{a,c\}</m>. Since the edge sets of the two graphs are not equal (as sets), the graphs are
         not equal (as graphs). </p>
     </solution>
@@ -724,7 +723,7 @@
     <solution>
       <p> The graphs are NOT equal, since <m>\{a,d\}
         \in E_1</m> but <m>\{a,d\} \notin E_2</m>. However, since both graphs contain the same number
-        of vertices and same number of edges, they <em>might</em> be isomorphic (this is not enough in
+        of vertices and the same number of edges, they <em>might</em> be isomorphic (this is not enough in
         most cases, but it is a good start). </p>
 
       <p> We can try to build an isomorphism. How about we
@@ -825,10 +824,8 @@
     <idx><h>graph</h><h>isomorphism class</h>
     </idx> Sometimes we will talk about a graph
     with a special name (like <m>K_n</m> or the <em>Petersen graph</em>) or perhaps draw a graph without
-    any labels. In this case we are really referring to <em>all</em> graphs isomorphic to any copy of
-    that particular graph. A collection of isomorphic graphs is often called an <term>isomorphism class</term>
-    .
-    <fn> This is not unlike geometry, where we might have more than one copy of a particular
+    any labels. In this case, we are really referring to <em>all</em> graphs isomorphic to any copy of
+    that particular graph. A collection of isomorphic graphs is often called an <term>isomorphism class</term>.<fn> This is not unlike geometry, where we might have more than one copy of a particular
     triangle. There instead of <em>isomorphic</em> we say <em>congruent</em>. </fn></p>
 
   <p>
@@ -1027,7 +1024,7 @@
     Back to some basic graph theory definitions. Notice that all the graphs we have drawn above have
     the property that no pair of vertices is connected more than once, and no vertex is connected to
     itself. Graphs like these are sometimes called <term>simple</term>, although we will just call them <em>
-    graphs</em>. This is because our definition for a graph says that the edges form a set of
+    graphs</em>. This is because our definition of a graph says that the edges form a set of
     2-element subsets of the vertices. Remember that it doesn't make sense to say a set contains an
     element more than once. So no pair of vertices can be connected by an edge more than once. Also,
     since each edge must be a set containing two vertices, we cannot have a single vertex connected
@@ -1041,7 +1038,7 @@
   <idx><h>multiset</h><h>relation
     to multigraph</h>
     </idx> That said, there are times we want to consider double (or more)
-    edges and single edge loops. For example, the <q>graph</q> we drew for the Bridges of Königsberg
+    edges and single-edge loops. For example, the <q>graph</q> we drew for the Bridges of Königsberg
     problem had double edges because there really are two bridges connecting a particular island to
     the near shore. We will call these objects <term>multigraphs</term>. This is a good name: a <em>multiset</em>
     is a set in which we are allowed to include a single element multiple times. </p>

source/sec_gt-trees.ptx

@@ -136,7 +136,7 @@
 
     <p>
       So far so good,
-      but while your grandparents are (probably) not blood-relatives,
+      but while your grandparents are (probably) not blood relatives,
       if we go back far enough,
       it is likely that they did have
       <em>some</em> common ancestor.
@@ -196,7 +196,7 @@
               </latex-image>
             </image>
             <p>
-              We are going to add edges to this graph to following some requirements.
+              We are going to add edges to this graph following some requirements.
             </p>
           </introduction>
           <task>
@@ -539,7 +539,7 @@
       Under each chapter, draw a vertex for each section,
       connecting it to the chapter it belongs to.
       The graph will not have any cycles; it will be a tree.
-      But a tree with clear hierarchy which is not present if we don't identify the book vertex as the <q>top</q>.
+      But a tree with a clear hierarchy which is not present if we don't identify the <q>book vertex</q> as the <q>top</q>.
     </p>
 
     <p>
@@ -556,7 +556,7 @@
     <p>
       <idx><h>descendant (in a rooted tree)</h></idx>
       <idx><h>ancestor (in a rooted tree)</h></idx>
-      Not surprisingly, the child of a child of a vertex is called the <term>grandchild</term> of the vertex (and it is the <term>grandparent</term>).  More in general, we say that a vertex <m>v</m> is a <term>descendent</term> of a vertex <m>u</m> provided <m>u</m> is a vertex on the path from <m>v</m> to the root.  Then we would call <m>u</m> an <term>ancestor</term> of <m>v</m>.
+      Not surprisingly, the child of a child of a vertex is called the <term>grandchild</term> of the vertex (and it is the <term>grandparent</term>).  More generally, we say that a vertex <m>v</m> is a <term>descendent</term> of a vertex <m>u</m> provided <m>u</m> is a vertex on the path from <m>v</m> to the root.  Then we would call <m>u</m> an <term>ancestor</term> of <m>v</m>.
     </p>
 
     <p>
@@ -630,15 +630,15 @@
     </example>
 
     <p>
-      <idx><h>breadth first search</h></idx>
-      <idx><h>search</h><h>breadth first</h></idx>
-      The key to how we partitioned the tree in the example was to know which vertex to assign to a set next.  We chose to visit all vertices in the same generation before any vertices of the next generation.  This is usually called a <term>breadth first search</term> (we say <q>search</q> because you often traverse a tree looking for vertices with certain properties).  
+      <idx><h>breadth-first search</h></idx>
+      <idx><h>search</h><h>breadth-first</h></idx>
+      The key to how we partitioned the tree in the example was to know which vertex to assign to a set next.  We chose to visit all vertices in the same generation before any vertices of the next generation.  This is usually called a <term>breadth-first search</term> (we say <q>search</q> because you often traverse a tree looking for vertices with certain properties).  
     </p>
 
     <p>
-      <idx><h>depth first search</h></idx>
-      <idx><h>search</h><h>depth first</h></idx>
-      In contrast, we could also have partitioned the tree in a different order.  Start with the root, put it in <m>A</m>.  Then look for one child of the root to put in <m>B</m>.  Then find a child of that vertex, into <m>A</m>, and then find its child, into <m>B</m>, and so on.  When you get to a vertex with no children, retreat to its parent and see if the parent has any other children.  So we travel as far from the root as fast as possible, then backtrack until we can move forward again.  This is called <term>depth first search</term>.
+      <idx><h>depth-first search</h></idx>
+      <idx><h>search</h><h>depth-first</h></idx>
+      In contrast, we could also have partitioned the tree in a different order.  Start with the root, put it in <m>A</m>.  Then look for one child of the root to put in <m>B</m>.  Then find a child of that vertex, into <m>A</m>, and then find its child, into <m>B</m>, and so on.  When you get to a vertex with no children, retreat to its parent and see if the parent has any other children.  So we travel as far from the root as fast as possible, then backtrack until we can move forward again.  This is called <term>depth-first search</term>.
     </p>
 
     <p>