diff --git a/chapter_1/1_2/1_2_3.md b/chapter_1/1_2/1_2_3.md new file mode 100644 index 0000000..a4b4e0a --- /dev/null +++ b/chapter_1/1_2/1_2_3.md @@ -0,0 +1,23 @@ +# 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. diff --git a/chapter_1/1_2/1_2_4.md b/chapter_1/1_2/1_2_4.md new file mode 100644 index 0000000..8b42be6 --- /dev/null +++ b/chapter_1/1_2/1_2_4.md @@ -0,0 +1,30 @@ +# 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. diff --git a/chapter_1/1_2/additional_exercises_1_2_6.md b/chapter_1/1_2/additional_exercises_1_2_6.md new file mode 100644 index 0000000..be77349 --- /dev/null +++ b/chapter_1/1_2/additional_exercises_1_2_6.md @@ -0,0 +1,378 @@ +# 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: + +$$ P \subseteq S $$ diff --git a/chapter_1/1_2/definition_1_2_1.md b/chapter_1/1_2/definition_1_2_1.md new file mode 100644 index 0000000..6ea8435 --- /dev/null +++ b/chapter_1/1_2/definition_1_2_1.md @@ -0,0 +1,15 @@ +# 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. diff --git a/chapter_1/1_2/definition_1_2_13.md b/chapter_1/1_2/definition_1_2_13.md new file mode 100644 index 0000000..22828e7 --- /dev/null +++ b/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 $P \leftrightarrow Q$. diff --git a/chapter_1/1_2/definition_1_2_6.md b/chapter_1/1_2/definition_1_2_6.md new file mode 100644 index 0000000..a3aaf7b --- /dev/null +++ b/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, $\neg P \to \neg Q$. diff --git a/chapter_1/1_2/definition_quantifiers_and_the_converse_contrapositive_inverse.md b/chapter_1/1_2/definition_quantifiers_and_the_converse_contrapositive_inverse.md new file mode 100644 index 0000000..ea51ff1 --- /dev/null +++ b/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** $\forall x \left(\neg P(x) \to \neg Q(x)\right)$ diff --git a/chapter_1/1_2/discrete_math_converse_contrapositive_inverse.png b/chapter_1/1_2/discrete_math_converse_contrapositive_inverse.png new file mode 100644 index 0000000..f1879eb Binary files /dev/null and b/chapter_1/1_2/discrete_math_converse_contrapositive_inverse.png differ diff --git a/chapter_1/1_2/discrete_math_necessary_and_sufficient.png b/chapter_1/1_2/discrete_math_necessary_and_sufficient.png new file mode 100644 index 0000000..c6c4232 Binary files /dev/null and b/chapter_1/1_2/discrete_math_necessary_and_sufficient.png differ diff --git a/chapter_1/1_2/discrete_math_quantifiers_converse_contrapositive_inverse.png b/chapter_1/1_2/discrete_math_quantifiers_converse_contrapositive_inverse.png new file mode 100644 index 0000000..0e5d28c Binary files /dev/null and b/chapter_1/1_2/discrete_math_quantifiers_converse_contrapositive_inverse.png differ diff --git a/chapter_1/1_2/discrete_math_truth_tables.png b/chapter_1/1_2/discrete_math_truth_tables.png new file mode 100644 index 0000000..34b35dc Binary files /dev/null and b/chapter_1/1_2/discrete_math_truth_tables.png differ diff --git a/chapter_1/1_2/investigate.md b/chapter_1/1_2/investigate.md new file mode 100644 index 0000000..e4ed1c6 --- /dev/null +++ b/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!! diff --git a/chapter_1/1_2/practice_problems_1_2_5.md b/chapter_1/1_2/practice_problems_1_2_5.md new file mode 100644 index 0000000..7969e09 --- /dev/null +++ b/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. diff --git a/chapter_1/1_2/preview_activity.md b/chapter_1/1_2/preview_activity.md new file mode 100644 index 0000000..86656dd --- /dev/null +++ b/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. diff --git a/chapter_1/1_2/reading_questions_1_2_4.md b/chapter_1/1_2/reading_questions_1_2_4.md new file mode 100644 index 0000000..6c5ad94 --- /dev/null +++ b/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). diff --git a/leftoff.txt b/leftoff.txt index 21e72e8..900731f 100644 --- a/leftoff.txt +++ b/leftoff.txt @@ -1 +1 @@ -48 +64