more typos

Author: oscarlevin

source/ch_logic.ptx

@@ -14,7 +14,7 @@
     </p>
 
     <p>
-      The problem is, as you no doubt know from arguing with friends, not all arguments are <em>good</em> arguments. A bad argument is one in which the conclusion does not follow from the premises, i.e., the conclusion is not a consequence of the premises. Logic is the study of what makes an argument good or bad. In other words, logic aims to determine in which cases a conclusion is, or is not, a consequence of a set of premises.
+      The problem is, as you no doubt know from arguing with friends, not all arguments are <em>good</em> arguments. A bad argument is one in which the conclusion does not follow from the premises; i.e., the conclusion is not a consequence of the premises. Logic is the study of what makes an argument good or bad. In other words, logic aims to determine in which cases a conclusion is, or is not, a consequence of a set of premises.
     </p>
 
 <!-- TODO: revise summary when settled on new structure -->

source/exercises/logic-implications.ptx

@@ -75,7 +75,7 @@
 
   <exercise>
     <statement>
-      <p> Write each of the following statements in the form, <q> If <ellipsis />, then <ellipsis />.</q> Careful, some statements may be false (which is alright for the purposes of this question). <ol>
+      <p> Write each of the following statements in the form, <q> If <ellipsis />, then <ellipsis />.</q> Careful, some statements may be false (which is fine for the purposes of this question). <ol>
           <li>
             <p> To lose weight, you must exercise. </p>
           </li>

source/exercises/logic-statements.ptx

@@ -91,7 +91,7 @@
           </li>
 
           <li>
-            <p> There is prime number that is even. </p>
+            <p> There is a prime number that is even. </p>
           </li>
 
           <li>

source/practice/logic-statements.ptx

@@ -88,7 +88,7 @@
   <exercise label="rs-statements-quant-interpretation">
     <statement>
       <p>
-        Regardless of your beliefs of how many people can be fooled at various times,  what could you conclude if we reinterpret <m>P(x,y)</m> to mean <m>x \lt y</m> and only quantify over the natural numbers (so <m>\forall x</m> means <q>for all natural numbers</q> and <m>\exists x</m> means <q>there exists a natural number</q>)?  Select all of the following that apply.
+        Regardless of your beliefs of how many people can be fooled at various times,  what could you conclude if we reinterpret <m>P(x,y)</m> to mean <m>x \lt y</m> and only quantify over the natural numbers (so <m>\forall x</m> means <q>For all natural numbers,</q> and <m>\exists x</m> means <q>There exists a natural number</q>)?  Select all of the following that apply.
       </p>
     </statement>
     <choices>

source/sec_logic-implications.ptx

@@ -34,7 +34,7 @@
 		<investigation>
 			<statement>
 				<p>
-					Little Timmy's Mom tells him, <q>if you don't eat all your broccoli, then you will not get any ice cream.</q>  Of course, Timmy loves his ice cream, so he quickly eats all his broccoli (which actually tastes pretty good). 
+					Little Timmy's Mom tells him, <q>If you don't eat all your broccoli, then you will not get any ice cream.</q>  Of course, Timmy loves his ice cream, so he quickly eats all his broccoli (which actually tastes pretty good). 
 				</p>
 				<p>
 					After dinner, when Timmy asks for his ice cream, he is told no!  Does Timmy have a right to be upset?  Why or why not?
@@ -62,11 +62,11 @@
 				</p>
 			 </blockquote>
 			 <p>
-				Math is about making general claims, but a claim is rarely going to be true of absolutely <em>every</em> mathematical object.  The way we <em>restrict</em> our claims to a particular type of object is with an implication: <q>take any object you like, <em>if</em> it is of the right type, <em>then</em> this thing is true about it.</q>
+				Math is about making general claims, but a claim is rarely going to be true of absolutely <em>every</em> mathematical object.  The way we <em>restrict</em> our claims to a particular type of object is with an implication: <q>Take any object you like, <em>if</em> it is of the right type, <em>then</em> this thing is true about it.</q>
 			 </p>
 
 			 <p>
-				Similarly, as we saw in the <xref ref="subsec_logic-statements-quant" text="title"/> subsection, when we make claims like <q>every square is a rectangle,</q> we really have an implication: <q>if something is a square, then it is a rectangle.</q> 
+				Similarly, as we saw in the <xref ref="subsec_logic-statements-quant" text="title"/> subsection, when we make claims like <q>Every square is a rectangle,</q> we really have an implication: <q>If something is a square, then it is a rectangle.</q> 
 			 </p>
 
 			 <p>
@@ -119,7 +119,7 @@
 		</p>
 
 		<figure xml:id="fig-implication-tt">
-			<caption>The truth table for <m>P \imp Q</m></caption>
+			<caption>The truth table for <m>P \imp Q</m>.</caption>
 			<tabular halign="center">
 				<col right="minor"/> <col right="major"/> <col/>
 				<row bottom="minor">
@@ -160,7 +160,7 @@
 		<exercise label="pa-sec-logic-implications-tommy">
 			<statement>
 				<p>
-					Consider the statement <q>If Tommy doesn't eat his broccoli, then he will not get any ice cream.</q>  Which of the following statements mean the same thing (i.e., will be true in the same situations)?  Select all that apply.
+					Consider the statement, <q>If Tommy doesn't eat his broccoli, then he will not get any ice cream.</q>  Which of the following statements mean the same thing (i.e., will be true in the same situations)?  Select all that apply.
 				</p>
 			</statement>
 			<choices randomize="yes" multiple-correct="yes">
@@ -268,7 +268,7 @@
 	<exercise label="pa-sec-logic-implications-quant">
 		<statement>
 			<p>
-				Consider the <em>sentence</em>, <q>if <m>x \ge 10</m>, then <m>x^2 \ge 25</m>.</q>  This sentence becomes a statement when we replace <m>x</m> by a value, or <q>capture</q> the <m>x</m> in the scope of a quantifier.  Which of the following claims are true (select all that apply)?
+				Consider the <em>sentence</em>, <q>If <m>x \ge 10</m>, then <m>x^2 \ge 25</m>.</q>  This sentence becomes a statement when we replace <m>x</m> by a value, or <q>capture</q> the <m>x</m> in the scope of a quantifier.  Which of the following claims are true (select all that apply)?
 			</p>
 		</statement>
 		<choices>
@@ -428,7 +428,7 @@
 					Then I have not lied; my statement is true.
 					However, if Bob did get a 90 on the final and did not pass the class,
 					then I lied, making the statement false.
-					The tricky case is this: what if Bob did not get a 90 on the final?
+					The tricky case is this: What if Bob did not get a 90 on the final?
 					Maybe he passes the class, maybe he doesn't.
 					Did I lie in either case?
 					I think not.
@@ -532,7 +532,7 @@
 		<example>
 			<statement>
 				<p>
-					Consider the statement, <q>all squares are rectangles,</q> which can also be phrased as, <q>for all shapes, if the shape is a square, then it is a rectangle.</q>  Is this statement true or false?  Are we sure?  What about the following three shapes?
+					Consider the statement, <q>All squares are rectangles,</q> which can also be phrased as, <q>For all shapes, if the shape is a square, then it is a rectangle.</q>  Is this statement true or false?  Are we sure?  What about the following three shapes?
 				</p>
 				<image width="75%">
 					<shortdescription>three shapes, a square, a non-square rectangle, and a triangle.</shortdescription>
@@ -555,15 +555,15 @@
 				</p>
 
 				<p>
-					Is the implication true of the rectangle in the middle?  Well, that shape is not a square (<m>P</m> is false) and it is a rectangle (<m>Q</m> is true).  But look, we believe that all squares are rectangles, so the statement must be true.  Even of a rectangle.  The only way this works is if <q>true implies false</q> is true!
+					Is the implication true of the rectangle in the middle?  Well, that shape is not a square (<m>P</m> is false), and it is a rectangle (<m>Q</m> is true).  But look, we believe that all squares are rectangles, so the statement must be true.  Even of a rectangle.  The only way this works is if <q>true implies false</q> is true!
 				</p>
 
 				<p>
-					Similarly, all squares are rectangles is a true statement, even when we look at a triangle.  <m>P</m> is false (the triangle is not a square) and <m>Q</m> is false (the triangle is not a rectangle).  Thankfully, we defined implications to be true in this case as well.
+					Similarly, all squares are rectangles is a true statement, even when we look at a triangle.  <m>P</m> is false (the triangle is not a square), and <m>Q</m> is false (the triangle is not a rectangle).  Thankfully, we defined implications to be true in this case as well.
 				</p>
 
 				<p>
-					We have given shapes that illustrate lines 1, 3, and 4 of the truth table for implications (<xref ref="fig-implication-tt"/>).  What shape illustrates line 2?  That would need to be a shape that was a square and was not a rectangle... Of course we can't find one, precisely because the statement is true!
+					We have given shapes that illustrate lines 1, 3, and 4 of the truth table for implications (<xref ref="fig-implication-tt"/>).  What shape illustrates line 2?  That would need to be a shape that was a square and was not a rectangle.... Of course we can't find one, precisely because the statement is true!
 				</p>
 			</solution>
 		</example>
@@ -661,13 +661,13 @@
 
 		<note>
 			<p>
-				It is unlikely that we would encounter a statement of the form <m>\exists x (P(x) \imp Q(x))</m>, since this would be automatically true if there was any <m>x</m> that made <m>P(x)</m> false.  But if we did, the same rules would apply to the converse, contrapositive, and inverse as above: just ignore the quantifier when swapping and/or negating the parts of the implication.
+				It is unlikely that we would encounter a statement of the form <m>\exists x (P(x) \imp Q(x))</m>, since this would be automatically true if there was any <m>x</m> that made <m>P(x)</m> false.  But if we did, the same rules would apply to the converse, contrapositive, and inverse as above: Just ignore the quantifier when swapping and/or negating the parts of the implication.
 			</p>
 		</note>
 
 
 		<p>
-			For example, <q>for all shapes, if the shape is a square, then it is a rectangle</q> (i.e., all squares are rectangles) has the converse, <q>for all shapes, if the shape is a rectangle, then it is a square</q> (so all rectangles are squares).
+			For example, <q>For all shapes, if the shape is a square, then it is a rectangle</q> (i.e., all squares are rectangles) has the converse, <q>For all shapes, if the shape is a rectangle, then it is a square</q> (so all rectangles are squares).
 		</p>
 
 		<p>
@@ -681,7 +681,7 @@
 		</p>
 
 		<p>
-			The contrapositive of <q>for all shapes, if it is a square, then it is a rectangle</q> is <q>for all shapes, if the shape is not a rectangle, then it is not a square.</q>  This is true.  In fact, <alert>the contrapositive of a true statement is always true</alert>!  
+			The contrapositive of <q>For all shapes, if it is a square, then it is a rectangle</q> is <q>For all shapes, if the shape is not a rectangle, then it is not a square.</q>  This is true.  In fact, <alert>the contrapositive of a true statement is always true</alert>!  
 		</p>
 
 		<p>
@@ -817,7 +817,7 @@
 							<p>
 								What would happen if Sue did not get an A but <em>did</em>
 								get a 93% on the final?
-								Then <m>P</m> would be true and <m>Q</m> would be false.
+								Then <m>P</m> would be true, and <m>Q</m> would be false.
 								This makes the implication <m>P \imp Q</m> false!
 								It must be that Sue did not get a 93% on the final.
 								Notice we now have the implication
@@ -885,12 +885,12 @@
 					(the <q>if</q> part)
 					<em>false</em>?
 					When I am in the shower but not singing.
-					That is the same condition for being false as the statement
-					<q>if I'm in the shower,
+					That is the same condition for being false as the statement,
+					<q>If I'm in the shower,
 					then I sing.</q> So the <q>if</q> part is <m>Q \imp P</m>.
 					On the other hand, to say,
 					<q>I sing only if I'm in the shower</q>
-					is equivalent to saying <q>if I sing,
+					is equivalent to saying <q>If I sing,
 					then I'm in the shower,</q> so the
 					<q>only if</q> part is <m>P \imp Q</m>.
 				</p>
@@ -901,14 +901,14 @@
 			It is not especially important to know which part is the
 			<q>if</q> or <q>only if</q> part,
 			but this does illustrate something very, very important:
-			<em>there are many ways to state an implication!</em>
+			<em>There are many ways to state an implication!</em>
 		</p>
 
 		<example>
 			<statement>
 				<p>
 					Rephrase the implication,
-					<q>if I dream, then I am asleep</q>
+					<q>If I dream, then I am asleep</q>
 					in as many ways as possible.
 					Then do the same for the converse.
 				</p>