fix typos

Author: oscarlevin

source/exercises/gt-conc.ptx

@@ -549,7 +549,7 @@
       <p>
         The graph <m>G</m> has 6 vertices with degrees <m>1, 2, 2, 3, 3, 5</m>.
         How many edges does <m>G</m> have?
-        If <m>G</m> was planar how many faces would it have?
+        If <m>G</m> was planar, how many faces would it have?
         Does <m>G</m> have an Euler trail?
       </p>
     </statement>
@@ -956,7 +956,7 @@
   <exercise>
     <statement>
       <p>
-        Consider the statement <q>If a graph is planar,
+        Consider the statement, <q>If a graph is planar,
         then it has an Euler trail.</q>
 
         <ol>
@@ -1052,7 +1052,7 @@
       </p>
 
       <p>
-        Note: this is asking you to prove a special case of Euler's formula for planar graphs, so do not use that formula in your proof.
+        Note: This is asking you to prove a special case of Euler's formula for planar graphs, so do not use that formula in your proof.
       </p>
     </statement>
     <hint>

source/sec_gt-conc.ptx

@@ -3,22 +3,22 @@
 <section xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="sec_gt-conc">
   <title>Chapter Summary</title>
   <p>
-    Hopefully this chapter has given you some sense for the wide variety of graph theory topics as well as why these studies are interesting.
-    There are many more interesting areas to consider and the list is increasing all the time;
+    Hopefully this chapter has given you some sense of the wide variety of graph theory topics as well as why these studies are interesting.
+    There are many more interesting areas to consider, and the list is increasing all the time;
     graph theory is an active area of mathematical research.
   </p>
 
   <p>
     One reason graph theory is such a rich area of study is that it deals with such a fundamental concept:
-    any pair of objects can either be related or not related.
-    What the objects are and what <q>related</q> means varies on context,
+    Any pair of objects can either be related or not related.
+    What the objects are and what <q>related</q> means varies depending on context,
     and this leads to many applications of graph theory to science and other areas of math.
     The objects can be countries,
     and two countries can be related if they share a border.
     The objects could be land masses that are related if there is a bridge between them.
     The objects could be websites that are related if there is a link from one to the other.
     Or we can be completely abstract:
-    the objects are vertices that are related if their is an edge between them.
+    The objects are vertices that are related if there is an edge between them.
   </p>
 
   <p>

source/sec_gt-matchings.ptx

@@ -74,9 +74,9 @@
     is a subset of the edges for which each vertex of <m>A</m> belongs to exactly one edge of the subset,
     and no vertex in <m>B</m> belongs to more than one edge in the subset.
     In practice, we will assume that <m>|A| = |B|</m>
-    (the two sets have the same number of vertices)
+    (the two sets have the same number of vertices),
     so this says that every vertex in the graph belongs to exactly one edge in the matching.<fn>
-    Note: what we are calling a <em>matching</em>
+    What we are calling a <em>matching</em>
     is sometimes called a <em>perfect matching</em>
     or <em>complete matching</em>.
     This is because it is interesting to look at non-perfect matchings as well.
@@ -240,7 +240,7 @@
       <p>
         We will have a matching if the matching condition holds.
         Given any set of card values (a set
-        <m>S \subseteq A</m>) we must show that <m>|N(S)| \ge |S|</m>.
+        <m>S \subseteq A</m>), we must show that <m>|N(S)| \ge |S|</m>.
         That is, the number of piles that contain those values is at least the number of different values.
         But what if it wasn't?
         Say <m>|S| = k</m>.