final touchups

Author: oscarlevin

cover/fibonacci_pascal.svg

@@ -2,6 +2,8 @@
 
 <chapter xmlns:xi="http://www.w3.org/2001/XInclude" xml:id="ch_logic">
   <title>Logic and Proofs</title>
+  <idx><h>self reference</h><see>reference, self</see></idx>
+  <idx><h>reference, self</h><see>self reference</see></idx>
   <introduction>
     <p>
       Logic is the study of consequence. Given a few mathematical statements or facts, we would like to be able to draw some conclusions. For example, if I told you that a particular real-valued function was continuous on the interval <m>[0,1]</m>, and <m>f(0) = -1</m> and <m>f(1) = 5</m>, can we conclude that there is some point between <m>[0,1]</m> where the graph of the function crosses the <m>x</m>-axis? Yes, we can, thanks to the Intermediate Value Theorem from calculus. Can we conclude that there is exactly one point? No. Whenever we find an <q>answer</q> in math, we really have a (perhaps hidden) argument. 
@@ -23,6 +25,7 @@
 
       Finally, we will put all of this together in <xref ref="sec_logic-proofs"/> and <xref ref="sec_logic-structures"/> to see how we can use these tools to construct arguments and prove statements.
     </p>
+
   </introduction>
   <xi:include href="./sec_logic-statements.ptx"/>
   <xi:include href="./sec_logic-implications.ptx" />

generated-assets/.crc-print_assets.pkl

@@ -61,13 +61,13 @@
   <exercise>
     <statement>
       <p>
-        Which of the graphs in the previous question contain Euler paths or circuits?
+        Which of the graphs in the previous question contain Euler trails or circuits?
         Which of the graphs are planar?
       </p>
     </statement>
     <solution>
       <p>
-        The first (and third) graphs contain an Euler path.
+        The first (and third) graphs contain an Euler trail.
         All the graphs are planar.
       </p>
     </solution>
@@ -89,7 +89,7 @@
   <exercise>
     <statement>
       <p>
-        Draw a graph that does not have an Euler path and is also not planar.
+        Draw a graph that does not have an Euler trail and is also not planar.
       </p>
     </statement>
     <solution>
@@ -168,7 +168,7 @@
 
           <li>
             <p>
-              Does <m>G</m> have an Euler path or circuit?
+              Does <m>G</m> have an Euler trail or circuit?
               Explain.
             </p>
           </li>
@@ -275,10 +275,10 @@
 
           <li>
             <p>
-              It is clear from the drawing that there is no Euler path,
+              It is clear from the drawing that there is no Euler trail,
               let alone an Euler circuit.
               Also, since there are more than 2 vertices of odd degree,
-              we know for sure there is no Euler path.
+              we know for sure there is no Euler trail.
             </p>
           </li>
         </ol>
@@ -318,7 +318,7 @@
 
           <li>
             <p>
-              For which values of <m>n</m> does the graph have an Euler path?
+              For which values of <m>n</m> does the graph have an Euler trail?
             </p>
           </li>
 
@@ -550,15 +550,15 @@
         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?
-        Does <m>G</m> have an Euler path?
+        Does <m>G</m> have an Euler trail?
       </p>
     </statement>
     <solution>
       <p>
         <m>G</m> has 8 edges
         (since the sum of the degrees is 16).
         If <m>G</m> is planar, then it will have 4 faces
-        (since <m>6 - 8 + 4 = 2</m>). <m>G</m> does not have an Euler path since there are more than 2 vertices of odd degree.
+        (since <m>6 - 8 + 4 = 2</m>). <m>G</m> does not have an Euler trail since there are more than 2 vertices of odd degree.
       </p>
     </solution>
   </exercise>
@@ -806,7 +806,7 @@
         <ol>
           <li>
             <p>
-              Does the graph have an Euler path or circuit?
+              Does the graph have an Euler trail or circuit?
               Explain.
             </p>
           </li>
@@ -839,7 +839,7 @@
         <ol>
           <li>
             <p>
-              The graph does have an Euler path, but not an Euler circuit.
+              The graph does have an Euler trail, but not an Euler circuit.
               There are exactly two vertices with odd degree.
               The path starts at one and ends at the other.
             </p>
@@ -893,7 +893,7 @@
 
           <li>
             <p>
-              Every bipartite graph has an Euler path.
+              Every bipartite graph has an Euler trail.
             </p>
           </li>
 
@@ -931,7 +931,7 @@
 
           <li>
             <p>
-              False. <m>K_{3,3}</m> has 6 vertices with degree 3, so contains no Euler path.
+              False. <m>K_{3,3}</m> has 6 vertices with degree 3, so contains no Euler trail.
             </p>
           </li>
 
@@ -957,7 +957,7 @@
     <statement>
       <p>
         Consider the statement <q>If a graph is planar,
-        then it has an Euler path.</q>
+        then it has an Euler trail.</q>
 
         <ol>
           <li>
@@ -1006,19 +1006,19 @@
         <ol>
           <li>
             <p>
-              If a graph has an Euler path, then it is planar.
+              If a graph has an Euler trail, then it is planar.
             </p>
           </li>
 
           <li>
             <p>
-              If a graph does not have an Euler path, then it is not planar.
+              If a graph does not have an Euler trail, then it is not planar.
             </p>
           </li>
 
           <li>
             <p>
-              There is a graph which is planar and does not have an Euler path.
+              There is a graph which is planar and does not have an Euler trail.
             </p>
           </li>
 
@@ -1031,13 +1031,13 @@
 
           <li>
             <p>
-              False. <m>K_4</m> is planar but does not have an Euler path.
+              False. <m>K_4</m> is planar but does not have an Euler trail.
             </p>
           </li>
 
           <li>
             <p>
-              False. <m>K_5</m> has an Euler path but is not planar.
+              False. <m>K_5</m> has an Euler trail but is not planar.
             </p>
           </li>
         </ol>

generated-assets/webwork/images/webwork-168-image-1.pdf

@@ -230,7 +230,7 @@
         <p>
           Consider the function <m>f:\N \to \N</m> given <em>recursively</em> by <me>
             f(0) = 1 \text{ and } f(n+1) = <var name="$a" /> \cdot f(n)
-          </me>. Find <me>f(<var name="$b" />)</me>.
+          </me>. Find <m>f(<var name="$b" />)</m>.
         </p>
         <p>
           <var name="$ans" width="25"/>

generated-assets/webwork/images/webwork-168-image-1.png

@@ -367,7 +367,7 @@
               How many strings have sum <m>n = 1</m>?
               How many have sum <m>n = 2</m>?
               And so on.
-              Find and explain a recurrence relation for the sequence <m>(a_n)</m> which gives the number of strings with sum <m>n</m>.
+              Find and explain a recurrence relation for the sequence <m>(a_n)</m> that gives the number of strings with sum <m>n</m>.
             </p>
           </li>
 

generated-assets/webwork/images/webwork-19-image-1.pdf

@@ -374,7 +374,7 @@
     </statement>
     <hint>
       <p>
-        This is a straight forward induction proof.
+        This is a straight-forward induction proof.
         Note you will need to simplify
         <m>\left(\frac{n(n+1)}{2}\right)^2 + (n+1)^3</m> and get <m>\left(\frac{(n+1)(n+2)}{2}\right)^2</m>.
       </p>

generated-assets/webwork/images/webwork-19-image-1.png

@@ -144,7 +144,7 @@
     <title>Telescoping to find a sum</title>
     <statement>
       <p>
-        Another context in which sequences arise is calculus when you study sequences and <term>series</term> (which is the word in calculus for what we call a sequences of partial sums).  Some of the techniques we have developed here can be applied there as well.  This is an example of a <term>telescoping sum</term>, similar to the telescoping technique we used.
+        Another context in which sequences arise is calculus when you study sequences and <term>series</term> (which is the word in calculus for what we call a sequence of partial sums).  Some of the techniques we have developed here can be applied there as well.  This is an example of a <term>telescoping sum</term>, similar to the telescoping technique we used.
       </p>
 
       <p>

generated-assets/webwork/images/webwork-35-image-1.pdf

@@ -519,7 +519,7 @@
         Now give a valid proof
         (by induction,
         even though you might be able to do so without using induction)
-        of the statement, <q>for all <m>n \in \N</m>,
+        of the statement, <q>For all <m>n \in \N</m>,
         the number <m>n^2 + n</m> is even.</q>
       </p>
     </statement>

generated-assets/webwork/images/webwork-35-image-1.png

@@ -5,7 +5,7 @@
   <exercise>
     <statement>
       <p>
-        Your friendly neighborhood bodega has a candy machine which gives 7 Skittles to the first customer who puts in a quarter, 10 to the second, 13 to the third, 16 to the fourth, etc.
+        Your friendly neighborhood bodega has a candy machine that gives 7 Skittles to the first customer who puts in a quarter, 10 to the second, 13 to the third, 16 to the fourth, etc.
         How many candies has the machine given out in total after 20 quarters are put into the machine?  After <m>n</m> quarters?
       </p>
     </statement>
@@ -14,7 +14,7 @@
   <exercise>
     <statement>
       <p>
-        Not to be outdone, the mega-mart across the street has installed a candy machine which gives 4 Skittles to the first customer, 7 to the second, 12 to the third, 19 to the fourth, etc.
+        Not to be outdone, the mega-mart across the street has installed a candy machine that gives 4 Skittles to the first customer, 7 to the second, 12 to the third, 19 to the fourth, etc.
         How many Skittles has the machine given out in total after 20 quarters are put into the machine?  After <m>n</m> quarters?
       </p>
     </statement>
@@ -234,7 +234,7 @@
       </sidebyside>
 
       <p>
-        The interesting thing here, is that a
+        The interesting thing here is that a
         <m>3\times 3</m> square now has area 13.
         Our goal is the find a formula for the area of a <m>n \times n</m> (diagonal) square.
 
@@ -254,7 +254,7 @@
             <p>
               Use your results from part (a) to find a closed formula for the sequence.
               Show your work.
-              Note, while there are lots of ways to find a closed formula here,
+              Note that while there are lots of ways to find a closed formula here,
               you should use partial sums specifically.
             </p>
           </li>