✏️ Lots of notes for very little progress

chapter_1/1_2/1_2_3.md

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+# Example 1.2.3
+
+Consider the statement:
+
+> If Bob gets a 90 on the final, then Bob will pass the class.
+
+This is definitely an implication: $P$ is the statement "Bob gets a 90 on the
+final," and $Q$ is the statement "Bob will pass the class."
+
+Suppose I made that statement to Bob. In what circumstances would it be fair to
+call me a liar? What if Bob really did get a 90 on the final, and he did pass
+the class? 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? Maybe
+he passes the class, maybe he doesn't. Did I lie in either case? I think not. In
+these last two cases, $P$ was false, and the statement $P \to Q$ was true. In
+the first case, $Q$ was true, and so was $P \to Q$. SO $P \to Q$ is true when
+$P$ is false or $Q$ is true.
+
+Student add on:
+
+Note that this example demonstrates that the implication is only false if $P$ is
+true an $Q$ is false.

chapter_1/1_2/1_2_4.md

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+# Example 1.2.4
+
+Decide which of the following statements are true and which are false. Briefly
+explain.
+
+1. If $1 = 1$, then most horses have 4 legs.
+
+2. If $0 = 1$, then $1 = 1$.
+
+3. If $8$ is a prime number, then the 7624th digit of $\pi$ is an $8$.
+
+4. If the 7624th digit of $\pi$ is an 8, then 2 + 2 = 4.
+
+**Solution**
+
+All four of the statements are true. Remember, the only way for an implication
+to be false is for the _if_ part to be true and the _then_ part to be false.
+
+1. Here both the hypothesis and the conclusion are true, so the implication is
+   true. It does not matter that there is no meaningful connection between the
+   true mathematical fact and the fact about horses.
+
+2. Here the hypothesis is false and the conclusion is true, so the implication
+   is true.
+
+3. I have no idea what the 7624th digit of $\pi$ is, but this does not matter.
+   Since the hypothesis is false, the implication is automatically true.
+
+4. Regardless of the truth value of the hypothesis, the conclusion is true,
+   making the implication true.

chapter_1/1_2/additional_exercises_1_2_6.md

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+# 1.2.6 Additional Exercises
+
+1.
+
+Q: Translate into English:
+
+(a) $\forall x (E(x) \to E(x + 2))$.
+
+(b) $\forall x \exists y (\sin(x) = y)$.
+
+\(c\) $\forall y \exists x (\sin(x) = y)$.
+
+(d) $\forall x \forall y \left(x^3 = y^3 \to x = y\right)$.
+
+A:
+
+(a) $\forall x (E(x) \to E(x + 2))$.
+
+For all numbers $x$, If $x$ is an even number, then it is true that $x + 2$ is
+an even number.
+
+(b) $\forall x \exists y (\sin(x) = y)$.
+
+For all numbers $x$, there exists at least one number $y$ where $\sin(x) = y$.
+
+\(c\) $\forall y \exists x (\sin(x) = y)$.
+
+For all numbers $y$, there exits at least one number $x$ where $\sin(x) = y$.
+
+(d) $\forall x \forall y \left(x^3 = y^3 \to x = y\right)$.
+
+For all numbers $x$ and for all numbers $y$, it is true that if $x^3 = y^3$,
+then it is true that $x = y$.
+
+2.
+
+Q: Consider the statement, "If Oscar eats Chinese food, then he drinks milk."
+
+(a) Write the converse statement.
+
+(b) Write the contrapositive of the statement.
+
+\(c\) Is it possible for the contrapositive to be false? If it was, what would
+that tell you?
+
+(d) Suppose the original statement is true, and that Oscar drinks milk. Can you
+conclude anything (about his eating Chinese food)? Explain.
+
+(e) Suppose the original statement is true, and that Oscar does not drink milk.
+Can you conclude anything (about his eating Chinese food)? Explain.
+
+A:
+
+Let $P$ be "Oscar eats Chinese food" and $Q$ be "Oscar drinks milk."
+
+$$ P \to Q $$
+
+Let's also review our statements:
+
+Given an implication $P \to Q$, we say,
+
+- The **converse** is the statement $Q \to P$.
+
+- The **contrapositive** is the statement $\neg Q \to \neg P$.
+
+- The **inverse** is the statement, $\neg P \to \neg Q$.
+
+(a) Write the converse statement.
+
+The converse statement is:
+
+$$ Q \to P $$
+
+In plain English it reads as: "If Oscar drinks milk, then he eats Chinese food."
+
+(b) Write the contrapositive of the statement.
+
+The contrapositive statement is $\neg Q \to \neg P$.
+
+In plain English it reads as: "If Oscar does not drink milk, then he does not
+eat Chinese food."
+
+\(c\) Is it possible for the contrapositive to be false? If it was, what would
+that tell you?
+
+Yes, it is possible for the contrapositive to be false, but then the original
+implication must also be false.
+
+(d) Suppose the original statement is true, and that Oscar drinks milk. Can you
+conclude anything (about his eating Chinese food)? Explain.
+
+Consider our truth tables:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| T   | F   | F         |
+| F   | T   | T         |
+| F   | F   | T         |
+
+Specifically where we know $Q$ is true:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| F   | T   | T         |
+
+We can see here that we cannot know whether Oscar ate Chinese food $P$, simply
+because we know that he drank milk $Q$, even though the statement $P \to Q$ is
+true in either case.
+
+(e) Suppose the original statement is true, and that Oscar does not drink milk.
+Can you conclude anything (about his eating Chinese food)? Explain.
+
+Again, let us consult our truth tables where $Q$ is false:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | F   | F         |
+| F   | F   | T         |
+
+But we know that $P \to Q$ is true as it is stated in the problem statement, so:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| F   | F   | T         |
+
+And so we _can_ conclude that if $P \to Q$ is true, and $Q$ is false, then we
+know that $P$ is false as well. In other words, we can conclude that Oscar did
+not eat Chinese food.
+
+3.
+
+Q: Write each of the following statements in the form, "If ..., then...."
+Careful, some statements may be false (which is fine for the purposes of this
+question).
+
+(a) To lose weight, you must exercise.
+
+(b) To lose weight, all you need to do is exercise.
+
+\(c\) Every American is patriotic.
+
+(d) You are patriotic only if you are American.
+
+(e) The set of rational numbers is a subset of the real numbers.
+
+(f) A number is prime if it is not even.
+
+(g) Either the Broncos will win the Super Bowl, or they won't play in the Super
+Bowl.
+
+A:
+
+(a) To lose weight, you must exercise.
+
+"If you are to lose weight, then it is necessary for you to exercise."
+
+(b) To lose weight, all you need to do is exercise.
+
+"If you exercise, then you will lose weight."
+
+\(c\) Every American is patriotic.
+
+"If you are an American, then you are patriotic."
+
+(d) You are patriotic only if you are American.
+
+"If you are patriotic, then you must be an American."
+
+(e) The set of rational numbers is a subset of the real numbers.
+
+"If a number is in the set of rational numbers, then it is in the set of real
+numbers."
+
+(f) A number is prime if it is not even.
+
+"If a number is prime, then it is not even."
+
+(g) Either the Broncos will win the Super Bowl, or they won't play in the Super
+Bowl.
+
+$$ P \vee \neg Q \equiv Q \to P $$
+
+"If the Broncos play in the Super Bowl, then the Broncos win the Super Bowl."
+
+4.
+
+Q: Consider the implication, "If you clean your room, then you can watch TV."
+Rephrase the implication in as many ways as possible. Then do the same for the
+converse.
+
+A:
+
+"If you clean your room, then you can watch TV."
+
+Rephrasing of original implication:
+
+1. You can watch TV if you clean your room.
+
+2. You clean your room only if you can watch TV.
+
+3. In order to watch TV, you must clean your room.
+
+4. To watch TV, it is necessary that you clean your room.
+
+5. To watch TV, it is sufficient to clean your room.
+
+6. You did not clean your room unless you can watch TV.
+
+And the converse:
+
+"If you can watch TV, then you clean your room."
+
+1. You clean your room if you can watch TV.
+
+2. It is necessary that you clean your room in order for you to watch TV.
+
+3. If you don't clean your room, then you cannot watch TV.
+
+---
+
+5.
+
+Q: Recall from calculus, if a function is differentiable at a point $c$, then it
+is continuous at $c$, but that the converse of this statement is not true (for
+example, $f(x) = |x|$ at the point 0). Restate this fact using "necessary and
+sufficient language."
+
+A:
+
+Let us recall:
+
+- "$P$ is necessary for $Q$" means $Q \to P$.
+
+- "$P$ is sufficient for $Q$" means $P \to Q$.
+
+- If $P$ is necessary and sufficient for $Q$, then $P \leftrightarrow Q$.
+
+Let $P$ be "A function is differentiable at a point $c$" and $Q$ be "the
+function is continuous at point $c$." Therefore we can say:
+
+- "$P$ is necessary for $Q$" means $Q \to P$ (this is false in this case).
+
+"Continuity is necessary but not sufficient for differentiability at $c$."
+
+- "$P$ is sufficient for $Q$" means $P \to Q$.
+
+"Differentiability is sufficient for continuity."
+
+- If $P$ is necessary and sufficient for $Q$, then $P \leftrightarrow Q$.
+
+"Continuity is necessary but not sufficient for differentiability at $c$."
+
+6.
+
+Q: Consider the statement, "For all natural numbers $n$, if $n$ is prime, then
+$n$ is solitary." You do not need to know what _solitary_ means for this
+problem, just that it is a property that some numbers have and others do not.
+
+(a) Write the converse and the contrapositive of the statement, saying which is
+which. Note: the original statement claims that an implication is true for all
+$n$, and it is that implication that we are taking the converse and
+contrapositive of.
+
+(b) Write the negation of the original statement. What would you need to show to
+prove that the statement is false?
+
+\(c\) Even though you don't know whether 10 is solitary (in fact, nobody knows
+this), is the statement, "If 10 is prime, then 10 is solitary" true or false?
+Explain.
+
+(d) It turns out that 8 is solitary. Does this tell you anything about the truth
+or fasilty of the original statement, its converse or its contrapositive?
+Explain.
+
+(e) Assuming that the original statement is true, what can you say about the
+relationship between the _set_ $P$ of prime numbers and the _set_ $S$ of
+solitary numbers. Explain.
+
+A:
+
+Let's first express this:
+
+$$ \forall n (P(n) \to S(n)) $$
+
+Let's also review our quantified converse/contrapositive definitions:
+
+A quantified implication $\forall x (P(x) to Q(x))$ has:
+
+**Converse** $\forall x(Q(x) \to P(x))$
+
+**Contrapositive** $\forall x \left(\neg Q(n) \to \neg P(x)\right)$
+
+(a) Write the converse and the contrapositive of the statement, saying which is
+which. Note: the original statement claims that an implication is true for all
+$n$, and it is that implication that we are taking the converse and
+contrapositive of.
+
+**Converse** $\forall n(S(n) \to P(n))$
+
+"For all natural numbers $n$, if $n$ is solitary, then $n$ is prime."
+
+**Contrapositive** $\forall n \left(\neg S(n) \to \neg P(n)\right)$
+
+"For all natural numbers $n$, if $n$ is not solitary, then $n$ is not prime."
+
+(b) Write the negation of the original statement. What would you need to show to
+prove that the statement is false?
+
+$$ \neg \forall n (P(n) \to S(n)) $$
+
+To prove that this statement is false, there would have to exist at least one
+value for $n$ where $n$ is prime and $n$ is not solitary.
+
+$$ \exists n \neg (P(n) \to S(n)) $$
+
+\(c\) Even though you don't know whether 10 is solitary (in fact, nobody knows
+this), is the statement, "If 10 is prime, then 10 is solitary" true or false?
+Explain.
+
+This falls within our truth table as:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| T   | F   | F         |
+
+We cannot know if 10 is solitary or not as stated in the problem statement. We
+only know that 10 is prime. So therefore we do not know if the statement
+$P \to Q$, or "If 10 is prime, then 10 is solitary", is true or not.
+
+(d) It turns out that 8 is solitary. Does this tell you anything about the truth
+or falsity of the original statement, its converse or its contrapositive?
+Explain.
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| F   | T   | T         |
+
+Yes, this tells us that since we know that 8 is solitary, it does not matter if
+8 is prime or not. Either way, "If 8 is prime, then 8 is solitary" ($P \to Q$),
+is a true statement.
+
+- The **converse** is the statement $Q \to P$.
+
+In this context this means "If 8 is solitary, then 8 is prime" ($Q \to P$), if
+we then invert the truth table, we get (where we know $Q$ is true):
+
+| $Q$ | $P$ | $Q \to P$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| T   | F   | F         |
+
+And we cannot verify the truth or falsity of $Q \to P$ nor the truth or falsity
+of $P$.
+
+- The **contrapositive** is the statement $\neg Q \to \neg P$.
+
+This must be true if the original implication is true. This is a true statement.
+
+(e) Assuming that the original statement is true, what can you say about the
+relationship between the _set_ $P$ of prime numbers and the _set_ $S$ of
+solitary numbers. Explain.
+
+We are given the statement:
+
+$$ \forall n (P(n) \to S(n)) $$
+
+This says that for every natural number $n$, if $n$ is prime, then $n$ is
+solitary. In other words, every prime number has the property of being solitary.
+
+We can interpret this in terms of sets as saying that every element of the set
+of prime numbers is also an element of the set of solitary numbers. Therefore,
+the set of prime numbers is a subset of the set of solitary numbers:
+
+$$ P \subseteq S $$

chapter_1/1_2/definition_1_2_1.md

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+# Definition 1.2.1 Implication
+
+An **implication** (or **conditional**) is a molecular statement of the form
+
+$$ P \to Q $$
+
+where $P$ and $Q$ are statements. We say that
+
+- $P$ is the **hypothesis** (or **antecedent**).
+
+- $Q$ is the **conclusion** (or **consequent**).
+
+An implication is _true_ provided $P$ is false or $Q$ is true (or both), and
+_false_ otherwise. In particular, the only way for $P \to Q$ to be false is for
+$P$ to be true _and_ $Q$ to be false.

chapter_1/1_2/definition_1_2_13.md

@@ -0,0 +1,7 @@
+# Definition 1.2.13 Necessary and Sufficient.
+
+- "$P$ is necessary for $Q$" means $Q \to P$.
+
+- "$P$ is sufficient for $Q$" means $P \to Q$.
+
+- If $P$ is necessary and sufficient for $Q$, then $P \leftrightarrow Q$.

chapter_1/1_2/definition_1_2_6.md

@@ -0,0 +1,9 @@
+# Definition 1.2.6 Converse, Contrapositive, and Inverse.
+
+Given an implication $P \to Q$, we say,
+
+- The **converse** is the statement $Q \to P$.
+
+- The **contrapositive** is the statement $\neg Q \to \neg P$.
+
+- The **inverse** is the statement, $\neg P \to \neg Q$.

chapter_1/1_2/definition_quantifiers_and_the_converse_contrapositive_inverse.md

@@ -0,0 +1,9 @@
+# Quantifiers and the Converse, Contrapositive, and Inverse.
+
+A quantified implication $\forall x (P(x) to Q(x))$ has:
+
+**Converse** $\forall x(Q(x) \to P(x))$
+
+**Contrapositive** $\forall x \left(\neg Q(x) \to \neg P(x)\right)$
+
+**Inverse** $\forall x \left(\neg P(x) \to \neg Q(x)\right)$

chapter_1/1_2/investigate.md

@@ -0,0 +1,63 @@
+# Investigate!
+
+Q: Little Timmy's Mom tells him, "If you don't eat all your broccoli, then you
+will not get any ice cream." Of course, Timmy loves his ice cream, so he quickly
+eats all his broccoli (which actually tastes pretty good).
+
+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?
+
+A: Well, probably, but in the context of this class, I'm guessning no?
+
+Let's think about this logically:
+
+Let $P$ be the predicate "If you don't eat all your broccoli", and let $Q$ be
+the conclusion "you will not get any ice cream."
+
+This is expressed as:
+
+$$ P \to Q $$
+
+If we consult our truth tables from the last section, we have:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| T   | F   | F         |
+| F   | T   | T         |
+| F   | F   | T         |
+
+But remember that Timmy did eat his broccoli, so that is actually $\neg P$.
+
+| $P$ | $\neg P$ |
+| --- | -------- |
+| T   | F        |
+| F   | T        |
+
+This changes our if/then truth table:
+
+| $\neg P$ | $Q$ | $P \to Q$ |
+| -------- | --- | --------- |
+| F        | T   | T         |
+| F        | F   | F         |
+| T        | T   | T         |
+| T        | F   | T         |
+
+Of note is the last three columns. The first of which shows that definitively,
+Timmy's Mom is correct, if Timmy doesn't eat his broccoli, he will definitely
+not get any ice cream:
+
+| $\neg P$ | $Q$ | $P \to Q$ |
+| -------- | --- | --------- |
+| F        | F   | F         |
+
+But notice the last two columns, where "If you eat all your broccoli" is true:
+
+| $\neg P$ | $Q$ | $P \to Q$ |
+| -------- | --- | --------- |
+| T        | T   | T         |
+| T        | F   | T         |
+
+This equates to The mom explicitly saying "If you do eat your broccoli, you may
+or may not get ice cream." Which is true, she never explicitly said that, lol.
+But hey, Timmy's Mom, come on man!!

chapter_1/1_2/practice_problems_1_2_5.md

@@ -0,0 +1,430 @@
+# 1.2.5 Practice Problems
+
+1.
+
+Q: In my safe is a sheet of paper with two shapes drawn on it in colored crayon.
+One is a circle, and the other is a pentagon. Each shape is drawn in a single
+color. Suppose you believe me when I tell you that, "If the circle is purple,
+then the pentagon is orange." What do you therefore know about the truth value
+of the following statements?
+
+(a) The circle and the pentagon are both purple.
+
+(b) The circle and the pentagon are both orange.
+
+\(c\) The circle is not purple, or the pentagon is orange.
+
+(d) If the pentagon is orange, then the circle is purple.
+
+(e) If the pentagon is not orange, then the circle is not purple.
+
+A:
+
+Let $P$ be "The circle is purple" and $Q$ be "The pentagon is orange."
+
+$$ P \to Q $$
+
+For reference let's also pull up our truth table:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| T   | F   | F         |
+| F   | T   | T         |
+| F   | F   | T         |
+
+(a) The circle and the pentagon are both purple.
+
+This follows the truth table of:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | F   | F         |
+
+So we therefore know that (a) is a false statement.
+
+(b) The circle and the pentagon are both orange.
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| F   | T   | T         |
+| F   | F   | T         |
+
+Note that we include both of these because since $P$ is false, it does not
+matter if $Q$ is true, we only know that $P \to Q$ is true.
+
+While $P \to Q$ is definitely true, we cannot determine if $Q$ is true or false,
+and therefore the validity of statement (b) is unknown.
+
+\(c\) The circle is not purple, or the pentagon is orange.
+
+This equates to:
+
+$$ \neg P \vee Q $$
+
+Let's "flip" our values for $P$ in our truth table to reflect this:
+
+| $\neg P$ | $Q$ | $P \to Q$ |
+| -------- | --- | --------- |
+| F        | T   | T         |
+| F        | F   | F         |
+| T        | T   | T         |
+| T        | F   | T         |
+
+And now isolate our statement in (c):
+
+| $\neg P$ | $Q$ | $P \to Q$ |
+| -------- | --- | --------- |
+| T        | T   | T         |
+| T        | F   | T         |
+
+This is a true statement $P \to Q$, as it tells us that the circle is not
+purple, and the pentagon _could_ be orange or not orange.
+
+(d) If the pentagon is orange, then the circle is purple.
+
+This is the converse statement to the implication.
+
+$$ Q \to P $$
+
+This is not true, we cannot know this to be true based off the implication. In
+other words, just because the pentagon is orange does not _necessarily_ mean
+that the circle is purple.
+
+The statement (d) is unknown.
+
+(e) If the pentagon is not orange, then the circle is not purple.
+
+This is the contrapositive:
+
+$$ \neg Q \to \neg P $$
+
+As we know, the contrapositive is always true if the implication is true. Since
+we established in the original problem statement that our implication is true,
+the contrapositive must also be true.
+
+2.
+
+Q: Suppose the statement, "_If the square is yellow, then the circle is purple_"
+is true. Assume also that the converse is false. Classify each statement below
+as true or false (if possible).
+
+(a) The circle is purple.
+
+(b) The square is yellow if and only if the circle is not purple.
+
+\(c\) The square is yellow.
+
+(d) The square is yellow if and only if the circle is purple.
+
+A:
+
+So let's start with our implication:
+
+Let $P$ be "The square is yellow", and the conclusion $Q$ be "The circle is
+purple."
+
+$$ P \to Q $$
+
+We also know that the converse is false:
+
+$$ \neg(Q \to P) $$
+
+And again, our truth tables:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| T   | F   | F         |
+| F   | T   | T         |
+| F   | F   | T         |
+
+(a) The circle is purple.
+
+Since this is the predicate of the converse statement, then the only thing we
+know to be true given our two assumptions is that the square is not yellow.
+Ultimately though, we do not know if this is a true statement or not, so it is
+not possible to classify this statement as either true or false.
+
+(b) The square is yellow if and only if the circle is not purple.
+
+This statement is saying:
+
+$$ P \leftrightarrow \neg Q $$
+
+But in order for this to be true, then both of the following statements would
+also have to be true:
+
+$$ P \to \neg Q $$
+
+"If the square is yellow, then the circle is not purple."
+
+$$ \neg Q \to P $$
+
+"If the circle is not purple, then the square is yellow."
+
+The first statement directly contradicts the implication and the second
+statement is not known.
+
+This statement is false due to the contradiction of the first statement with the
+implication.
+
+\(c\) The square is yellow.
+
+We cannot know if this statement is true or not. If it is, then we know that the
+circle is purple, but again, this statement cannot be classified as either true
+or false.
+
+(d) The square is yellow if and only if the circle is purple.
+
+This statement is saying:
+
+$$ P \leftrightarrow Q $$
+
+This is false, in order for this statement to be true, both the implication as
+well as its converse must be true, but the original problem statement tells us
+the implication is true and the converse is false.
+
+This statement is false.
+
+3.
+
+Q: Consider the statement, "_If you will give me magic beans, then I will give
+you a cow._" Decide whether each statement below is the converse, the
+contrapositive, or neither.
+
+(a) If I will give you a cow, then you will not give me magic beans.
+
+(b) If I will give you a cow, then you will give me magic beans.
+
+\(c\) If you will not give me magic beans, then I will not give you a cow.
+
+(d) If you will give me magic beans, then I will not give you a cow.
+
+(e) You will give me magic beans, then I will not give you a cow.
+
+(f) If I will not give you a cow, then you will not give me magic beans.
+
+A:
+
+Let $P$ be "You will give me magic beans" and $Q$ be "I will give you a cow."
+
+And let us review definition 1.2.6:
+
+Given an implication $P \to Q$, we say,
+
+- The **converse** is the statement $Q \to P$.
+
+- The **contrapositive** is the statement $\neg Q \to \neg P$.
+
+- The **inverse** is the statement, $\neg P \to \neg Q$.
+
+(a) If I will give you a cow, then you will not give me magic beans.
+
+This equates to:
+
+$$ Q \to \neg P $$
+
+This does not correspond to any of our definitions.
+
+(b) If I will give you a cow, then you will give me magic beans.
+
+This equates to:
+
+$$ Q \to P $$
+
+This is the **converse** statement.
+
+\(c\) If you will not give me magic beans, then I will not give you a cow.
+
+This equates to:
+
+$$ \neg P \to \neg Q $$
+
+This is the **inverse** statement.
+
+(d) If you will give me magic beans, then I will not give you a cow.
+
+This equates to:
+
+$$ P \to \neg Q $$
+
+This does not correspond to any of the definitions.
+
+(e) You will give me magic beans, then I will not give you a cow.
+
+This equates to:
+
+$$ P \to \neg Q $$
+
+Which again, does not correspond to any of the definitions.
+
+(f) If I will not give you a cow, then you will not give me magic beans.
+
+This equates to:
+
+$$ \neg Q \to \neg P $$
+
+This is the **contrapositive** statement.
+
+4.
+
+Q: You have discovered an old paper on graph theory that discusses the
+_viscosity_ of a graph (which for all you know, is something completely made up
+by the author). A theorem in the paper claims that "if a graph satisfies
+_condition_ _(V)_, then the graph is _viscous_." Which of the following are
+equivalent ways of stating this claim? Which are equivalent to the converse of
+the claim?
+
+(a) Only viscous graphs satisfy condition (V).
+
+(b) For a graph to be viscous, it is necessary that it satisfies condition (V).
+
+\(c\) A graph is viscous only if it satisfies condition (V).
+
+(d) Satisfying condition (V) is a necessary condition for a graph to be viscous.
+
+(e) A graph is viscous if it satisfies condition (V).
+
+A:
+
+Let $P$ be "A graph satisfies condition (V)" and $Q$ be "the graph is viscous."
+
+Let's also review the _converse_ definition:
+
+Given an implication $P \to Q$, we say,
+
+- The **converse** is the statement $Q \to P$.
+
+And also let's also review necessary/sufficient wording definitions:
+
+- "$P$ is necessary for $Q$" means $Q \to P$.
+
+- "$P$ is sufficient for $Q$" means $P \to Q$.
+
+- If $P$ is necessary and sufficient for $Q$, then $P \leftrightarrow Q$.
+
+(a) Only viscous graphs satisfy condition (V).
+
+This is:
+
+$$ Q \to P $$
+
+Which is equivalent to the _converse_ of the claim.
+
+(b) For a graph to be viscous, it is necessary that it satisfies condition (V).
+
+Recall the wording:
+
+- "$P$ is necessary for $Q$" means $Q \to P$.
+
+This is the _converse_ of the claim.
+
+\(c\) A graph is viscous only if it satisfies condition (V).
+
+- "$P$ is sufficient for $Q$" means $P \to Q$.
+
+"only if it satisfies" usually says something along the lines of "I am $Q$ only
+if I am $P$."
+
+But again, statement \(c\) is saying $Q$ only if $P$, so it is reversed, so this
+lines up as a **converse**.
+
+This is equivalent to the _converse_ of the claim.
+
+(d) Satisfying condition (V) is a necessary condition for a graph to be viscous.
+
+- "$P$ is necessary for $Q$" means $Q \to P$.
+
+And this is a direct wording to our (d) statement, so:
+
+This is equivalent to the _converse_ of the claim.
+
+(e) A graph is viscous if it satisfies condition (V).
+
+This is the same as the original implication, just reversed, looking for the
+"if" statement tells us which is $P$ and which is $Q$.
+
+$$ P \to Q $$
+
+This is an equivalent way of stating the original claim.
+
+5.
+
+Q: Which of the following statements are equivalent to the implication, "_if you
+win the lottery, then you will be rich,_" and which are equivalent to the
+converse of the implication?
+
+(a) If you are not rich, then you did not win the lottery.
+
+(b) It is sufficient to win the lottery to be rich.
+
+\(c\) Either you win the lottery, or else you are not rich.
+
+(d) If you are rich, you must have won the lottery.
+
+(e) You will win the lottery if and only if you are rich.
+
+A:
+
+Let $P$ be "You win the lottery" and $Q$ be "You will be rich".
+
+Let's also review the _converse_ definition:
+
+Given an implication $P \to Q$, we say,
+
+- The **converse** is the statement $Q \to P$.
+
+And also let's also review necessary/sufficient wording definitions:
+
+- "$P$ is necessary for $Q$" means $Q \to P$.
+
+- "$P$ is sufficient for $Q$" means $P \to Q$.
+
+- If $P$ is necessary and sufficient for $Q$, then $P \leftrightarrow Q$.
+
+(a) If you are not rich, then you did not win the lottery.
+
+This is:
+
+$$ \neg Q \to \neg P $$
+
+This is the _contrapositive_ of the implication statement, not the _converse_,
+but it is always true if the implication is true. Therefore they are equivalent
+statements.
+
+(b) It is sufficient to win the lottery to be rich.
+
+This is equivalent to the implication as:
+
+- "$P$ is sufficient for $Q$" means $P \to Q$.
+
+\(c\) Either you win the lottery, or else you are not rich.
+
+This is saying:
+
+$$ P \vee \neg Q $$
+
+This is equivalent to:
+
+$$ P \vee \neg Q \equiv Q \to P $$
+
+This is the _converse_ statement.
+
+(d) If you are rich, you must have won the lottery.
+
+This is:
+
+$$ Q \to P $$
+
+This is the _converse_ of the implication.
+
+(e) You will win the lottery if and only if you are rich.
+
+This is:
+
+$$ Q \leftrightarrow P $$
+
+This is a bidirectional statement, which will only be true if both the
+implication and its converse statement are true. It is not equivalent to either
+the implication nor its converse.

chapter_1/1_2/preview_activity.md

@@ -0,0 +1,345 @@
+# Preview Activity
+
+1.
+
+Q: Consider the statement, "If Tommy doesn't eat his broccoli, then he will not
+get any ice cream." Which of the following statements mean the same thing
+(_i.e._, will be true in the same situations)? Select all that apply.
+
+A. If Tommy does eat his broccoli, then he will get ice cream.
+
+B. If Tommy gets ice cream, then he ate his broccoli.
+
+C. If Tommy doesn't get ice cream, then he didn't eat his broccoli.
+
+D. Tommy ate his broccoli and still didn't get any ice cream.
+
+A:
+
+Let's first establish:
+
+$$ P = \text{Tommy doesn't eat his broccoli} $$
+
+$$ Q = \text{He will not get any ice cream} $$
+
+And also let's establish our truth tables as:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| T   | F   | F         |
+| F   | T   | T         |
+| F   | F   | T         |
+
+A. If Tommy does eat his broccoli, then he will get ice cream.
+
+This equates to the last row of the truth table above:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| F   | F   | T         |
+
+Or:
+
+$$ \neg P \wedge \neg Q \to (P \to Q) $$
+
+B. If Tommy gets ice cream, then he ate his broccoli.
+
+The two rows where $Q$ is false are:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | F   | F         |
+| F   | F   | T         |
+
+As we can see, we don't know if Tommy ate his broccoli just because he got ice
+cream.
+
+C. If Tommy doesn't get ice cream, then he didn't eat his broccoli.
+
+This is the case where $Q$ is true:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| F   | T   | T         |
+
+And again, we do not know if Tommy ate his broccoli just because he didn't get
+his ice cream (equivalent to part B).
+
+D. Tommy ate his broccoli and still didn't get any ice cream.
+
+These are all situations where $P$ is false:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| F   | T   | T         |
+| F   | F   | T         |
+
+And Tommy didn't get any ice cream, so:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| F   | T   | T         |
+
+2.
+
+Q: Suppose that your shady uncle offers you the following deal: If you loan him
+your car, then he will bring you tacos. In which of the following situations
+would it be fair to say that your uncle is a liar (_i.e._, that his statement
+was false)? Select all that apply.
+
+A. You loan him your car. He brings you tacos.
+
+B. You loan him your car. He never buys you tacos.
+
+C. You don't loan him your car. He still brings you tacos.
+
+D. You don't loan him your car. He never brings you tacos.
+
+A:
+
+Let's first establish:
+
+$$ P = \text{You loan your uncle your car} $$
+
+$$ Q = \text{your uncle brings you tacos} $$
+
+The assumption is:
+
+$$ P \to Q $$
+
+And our truth tables again are:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| T   | F   | F         |
+| F   | T   | T         |
+| F   | F   | T         |
+
+A. You loan him your car. He brings you tacos.
+
+So here, it would not be fair to say our uncle is a liar, because he fulfilled
+the first row of the truth table:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+
+Which makes sense. In essence, he made a promise and then delivered.
+
+B. You loan him your car. He never buys you tacos.
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | F   | F         |
+
+Here it makes sense to call our uncle a liar. We loaned him our car $P$, but he
+never gave us tacos $Q$, so the assertion $P \to Q$ is false, and hence our
+uncle is a liar.
+
+C. You don't loan him your car. He still brings you tacos.
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| F   | T   | T         |
+
+While in the regular world, we would still call our uncle a liar, in the context
+of discrete mathematics logic, we actually would say our uncle told us the truth
+here.
+
+While we never loaned our uncle our car, $P$, he still gave us tacos $Q$, and
+the truth table tells us that $P \to Q$ is still true.
+
+D. You don't loan him your car. He never brings you tacos.
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| F   | F   | T         |
+
+Well here it actually would make sense in the real world and in the mathematical
+context to say our uncle is not a liar. We never fulfilled our promise to loan
+him the car $P$, and he never gave us tacos $Q$, so $P \to Q$ makes sense both
+from a daily life logical standpoint and also from a discrete math one.
+
+3.
+
+Q: Consider the _sentence_, "If $x \geq 10$, then $x^2 \geq 25$." This sentence
+becomes a statement when we replace $x$ by a value, or "capture" the $x$ in the
+scope of a quantifier. Which of the following claims are true (select all that
+apply)?
+
+A. If we replace $x$ by $15$, then the resulting statement is true. (Note,
+$15^2 = 225$.)
+
+B. If we replace $x$ by $3$, then the resulting statement is true.
+
+C. If we replace $x$ by $6$, then the resulting statement is true.
+
+D. The universal generalization ("for all $x$, if $x \geq 10$, then
+$x^2 \geq 25$") is true.
+
+E. There is a number we could replace $x$ with that makes the statement false.
+
+A:
+
+Again, let's say that:
+
+$$ P(x) = x \geq 10 $$
+
+$$ Q(x) = x^2 \geq 25 $$
+
+And truth tables:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | T   | T         |
+| T   | F   | F         |
+| F   | T   | T         |
+| F   | F   | T         |
+
+A. If we replace $x$ by $15$, then the resulting statement is true. (Note,
+$15^2 = 225$.)
+
+Let's evaluate:
+
+$$ P(x) = x \geq 10 $$
+
+$$ Q(x) = x^2 \geq 25 $$
+
+$$ P(15) = 15 \geq 10 \Rightarrow \text{True} $$
+
+$$ Q(15) = 225 \geq 25 \Rightarrow \text{True} $$
+
+| $P(15)$ | $Q(15)$ | $P(15) \to Q(15)$ |
+| ------- | ------- | ----------------- |
+| T       | T       | T                 |
+
+This statement is true.
+
+B. If we replace $x$ by $3$, then the resulting statement is true.
+
+Let's evaluate:
+
+$$ P(x) = x \geq 10 $$
+
+$$ Q(x) = x^2 \geq 25 $$
+
+$$ P(3) = 3 \geq 10 \Rightarrow \text{False} $$
+
+$$ Q(3) = 9 \geq 25 \Rightarrow \text{False} $$
+
+| $P(3)$ | $Q(3)$ | $P(3) \to Q(3)$ |
+| ------ | ------ | --------------- |
+| F      | F      | T               |
+
+This statement is true.
+
+C. If we replace $x$ by $6$, then the resulting statement is true.
+
+Let's evaluate:
+
+$$ P(x) = x \geq 10 $$
+
+$$ Q(x) = x^2 \geq 25 $$
+
+$$ P(6) = 6 \geq 10 \Rightarrow \text{False} $$
+
+$$ Q(6) = 36 \geq 25 \Rightarrow \text{True} $$
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| F   | T   | T         |
+
+This statement is true.
+
+D. The universal generalization ("for all $x$, if $x \geq 10$, then
+$x^2 \geq 25$") is true.
+
+Since this is a universal generalization, this has nothing to do with $P(x)$ or
+$Q(x)$, unless we explicitly define it as such. Without those definitions, this
+is saying:
+
+$$ \forall x \left(x \geq 10 \to \left(x^2 \geq 25\right)\right) $$
+
+Given our previous definitions for $P(x)$ and $Q(x)$, we could rewrite this as:
+
+$$ \forall x (P(x) \to Q(x)) $$
+
+This is a true statement. Since $10^2 = 100$, and any larger values for $x$ is
+guaranteed to be larger than $25$.
+
+E. There is a number we could replace $x$ with that makes the statement false.
+
+The intuition from part D indicates that this would have to be false, but let's
+consider our truth table to be sure:
+
+| $P$ | $Q$ | $P \to Q$ |
+| --- | --- | --------- |
+| T   | F   | F         |
+
+The only way an "if/then" statement in this context can be false is if the
+hypothesis, $P$, is true, but the conclusion, $Q$, is false.
+
+So this would actually be saying:
+
+$$ \exists x \neg (P(x) \to Q(x)) $$
+
+But this is not true, as long as the hypothesis holds true, then $Q(x)$ will
+also hold true.
+
+This statement is false.
+
+4.
+
+Q: Consider the statement, "If I see a movie, then I eat popcorn" (which happens
+to be true). Based solely on your intuition of English, which of the following
+statements mean the same thing? Select all that apply.
+
+A. If I eat popcorn, then I see a movie.
+
+B. If I don't eat popcorn, then I don't see a movie.
+
+C. It is necessary that I eat popcorn when I see a movie.
+
+D. To see a movie, it is sufficient for me to eat popcorn.
+
+E. I only watch a movie if I eat popcorn.
+
+A:
+
+So in this question, we're asking to solely use our intuition of English...
+
+A. If I eat popcorn, then I see a movie.
+
+This isn't necessarily true, as the original statement only says that If I see a
+movie that I then eat popcorn. This statement is saying that if I eat popcorn,
+then I see a movie. The first statement does not imply the other (I could eat
+popcorn and then do anything else).
+
+B. If I don't eat popcorn, then I don't see a movie.
+
+This is true, as the first statement says that if I see a movie, then I eat
+popcorn. Therefore if I'm not eating popcorn, I could be doing anything at all,
+but I am definitely not seeing a movie.
+
+C. It is necessary that I eat popcorn when I see a movie.
+
+While it is true that "If I see a movie, then I eat popcorn." Nothing about the
+statement claims that it is _necessary_ that I eat popcorn when I see a movie,
+only that it is true that when I see a movie, that I then eat popcorn.
+
+In the context of discrete math though, this is equivalent to the original
+statement.
+
+D. To see a movie, it is sufficient for me to eat popcorn.
+
+There is nothing in the original statement, "If I see a movie, then I eat
+popcorn" that makes any claims about the _sufficiency_ for me to eat popcorn.
+
+E. I only watch a movie if I eat popcorn.
+
+This is equivalent to the original statement.
+
+C & E are the answers.

chapter_1/1_2/reading_questions_1_2_4.md

@@ -0,0 +1,78 @@
+# 1.2.4 Reading Questions
+
+1.
+
+Q: It happens to be true that all mammals have hair. Which of the following are
+also true?
+
+A. Having hair is a necessary condition for being a mammal.
+
+B. Having hair is a sufficient condition for being a mammal.
+
+C. If an animal doesn't have hair, then it is not a mammal.
+
+D. An animal is a mammal only if it has hair.
+
+A:
+
+Let $P$ be "The anime is a mammal", $Q$ be "the animal has hair".
+
+A. Having hair is a necessary condition for being a mammal.
+
+"$P$ is necessary for $Q$" means $Q \to P$. But be careful here, the first
+statement in this case is $Q$ a,d the second statement is $P$, so we reverse in
+the other direction.
+
+$$ P \to Q $$
+
+This is our original implication, so this is true.
+
+B. Having hair is a sufficient condition for being a mammal.
+
+$$ Q \to P $$
+
+This is the converse. We cannot know if this is true based off the original
+implication.
+
+C. If an animal doesn't have hair, then it is not a mammal.
+
+This is the contrapositive.
+
+$$ \neg Q \to \neg P $$
+
+This is true (contrapositives are always true if the implication is true).
+
+D. An animal is a mammal only if it has hair.
+
+Do not be confused here, this is not "if and only if", it is the original
+implication.
+
+$$ P \to Q $$
+
+Which is the same as the original implication, this is true.
+
+2.
+
+Q: Given an example of a _true_ implication (written out in words) that has a
+_false_ converse. Explain why your implication is true and why the converse is
+false.
+
+A:
+
+Consider the statement "If an animal is a frog, then it is an amphibian."
+
+Let $P$ be the predicate "The animal is a frog", and the conclusion be "The
+animal is an amphibian."
+
+This is a true statement, all frogs are amphibians.
+
+$$ P \to Q $$
+
+Now consider the converse:
+
+$$ Q \to P $$
+
+This would read as "If an animal is an amphibian, then it must be a frog."
+
+This is not a true statement, as evidenced by the various amphibians that are
+not frogs (newts being one example).

leftoff.txt

@@ -1 +1 @@
-48
+64