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chapter_3/exercises.md
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@ -671,3 +671,284 @@ d. $a < b \text{ and } c < d \Rightarrow ac < bd$
This is false. Say $a = -1$, $b = 2$, $c = -8$ and $d = 3$. This would make
$-1 < 2$ and $-8 < 3$, which is true, but then $(-1)(-8) < (2)(3)$ would be
$8 < 6$, which is false.
---
**Exercise Set 3.2**
Page 152
1. Which of the following is a negation for "All discrete mathematics students
are athletic"? More than one answer may be correct.
a. There is a discrete mathematics student who is nonathletic.
b. All discrete mathematics students are nonathletic.
c. There is an athletic person who is not a discrete mathematics student.
d. No discrete mathematics students are athletic.
e. Some discrete mathematics students are nonathletic.
f. No athletic people are discrete mathematics students.
2. Which of the following is a negation for "All dogs are loyal"? More than one
answer may be correct.
a. All dogs are disloyal.
b. No dogs are loyal.
c. Some dogs are disloyal.
d. Some dogs are loyal.
e. There is a disloyal animal that is not a dog.
f. There is a dog that is disloyal.
g. No animals that are not dogs are loyal.
h. Some animals that are not dogs are loyal.
3. Write the formal negation for each of the following statements.
a. $\forall$ string $s$, $s$ has at least one character.
b. $\forall$ computer $c$, $c$ has a CPU.
c. $\exists$ a movie $m$ such that $m$ is over 6 hours long.
d. $\exists$ a band $b$ such that $b$ has won at least 10 Grammy awards.
4. Write an informal negation for each of the following statements. Be careful
to avoid negations that are ambiguous.
a. All dogs are friendly.
b. All graphs are connected.
c. Some suspicions were substantiated.
d. Some estimates are accurate.
5. Write a negation for each of the following statements.
a. Every valid argument has a true conclusion.
b. All real numbers are positive, negative, or zero.
Write a negation for each statement in 6 and 7.
6.
a. Sets $A$ and $B$ do not have any points in common.
b. Towns $P$ and $Q$ are not connected by any road on the map.
7.
a. This vertex is not connected to any other vertex in the graph.
b. This number is not related to any even number.
8. Consider the statement "There are no simple solutions to life's problems."
Write an informal negation for the statement, and then write the statement
formally using quantifiers and variables.
Write a negation for each statement in 9 and 10.
9. $\forall$ real number $x$, if $x > 3$ then $x^2 > 9$.
10. $\forall$ computer program $P$, if $P$ compiles without error messages, then
$P$ is correct.
In each of 11-14 determine whether the proposed negation is correct. If it is
not, write a correct negation.
11.
_Statement:_ The sum of any two irrational numbers is irrational.
_Proposed negation:_ The sum of any two irrational numbers is rational.
12.
_Statement:_ The product of any irrational number and any rational number is
irrational.
_Proposed negation:_ The product of any irrational number and any rational
number is rational.
13.
_Statement:_ For every integer $n$, if $n^2$ is even then $n$ is even.
_Proposed negation:_ For every integer $n$, if $n^2$ is even then $n$ is not
even.
14.
_Statement:_ For all real numbers $x_1$ and $x_2$, if $x_1^2 = x_2^2$ then
$x_1 = x_2$.
_Proposed negation:_ For all real numbers $x_1$ and $x_2$, if $x_1^2 = x_2^2$
then $x_1 \neq x_2$.
15. Let $D = \{-48, -14, -8, 0, 1, 3, 16, 23, 26, 32, 36\}$. Determine which of
the following statements are true and which are false. Provide
counterexamples for the statements that are false.
a. $\forall x \in D, \text{ if } x \text{ is odd then } x > 0$.
b.
$\forall x \in D, \text{ if } x \text{ is less than } 0 \text{ then } x \text{ is even}$.
c. $\forall x \in D, \text{ if } x \text{ is even then } x \leq 0$.
d.
$\forall x \in D, \text{ if the ones digit of } x \text{ is } 2, \text{ then the tens digit is } 3 \text{ or } 4$.
e.
$\forall x \in D, \text{ if the ones digit of } x \text{ is } 6, \text{ then the tens digit is } 1 \text{ or } 2$.
In 16-23, write a negation for each statement.
16. $\forall$ real number $x$, if $x^2 \geq 1$ then $x > 0$.
17. $\forall$ integer $d$, if $\dfrac{6}{d}$ is an integer then $d = 3$.
18. $\forall x \in \mathbb{R}$, if $x(x + 1) > 0$ then $x > 0$ or $x < -1$.
19. $\forall x \in \mathbb{Z}$, if $n$ is prime then $n$ is odd or $n = 2$.
20. $\forall$ integers $a$, $b$, and $c$, if $a - b$ is even and $b - c$ is
even, then $a - c$ is even.
21. $\forall$ integer $n$, if $n$ is divisible by $6$, then $n$ is divisible by
$2$ and $n$ is divisible by $3$.
22. If the square of an integer is odd, then the integer is odd.
23. If a function is differentiable then it is continuous.
24. Rewrite the statements in each pair in if-then form and indicate the logical
relationship between them.
a.
All the children in Tom's family are female.
All the females in Tom's family are children.
b.
All the integers that are greater than 5 and end in 1, 3, 7, or 9 are prime.
All the integers that are greater than 5 and are prime end in 1, 3, 7, or 9.
25. Each of the following statements is true. In each case write the converse
statement, and give a counterexample showing that the converse is false.
a. If $n$ is any prime number that is greater than $2$, then $n + 1$ is even.
b. If $m$ is an odd integer, then $2m$ is even.
c. If two circles intersect in exactly two points, then they do not have a
common center.
In 26-33, for each statement in the referenced exercise write the
contrapositive, converse, and inverse. Indicate as best as you can which of
these statements are true and which are false. Give a counterexample for each
that is false.
26. Exercise 16
27. Exercise 17
28. Exercise 18
29. Exercise 19
30. Exercise 20
31. Exercise 21
32. Exercise 22
33. Exercise 23
34. Write the contrapositive for each of the following statements.
a. If $n$ is prime, then $n$ is not divisible by any prime number from $2$
through $\sqrt{n}$. (Assume that $n$ is a fixed integer.)
b. If $A$ and $B$ do not have any elements in common, then they are disjoint.
(Assume that $A$ and $B$ are fixed sets.)
35. Give an example to show that a universal conditional statement is not
logically equivalent to its inverse.
36. If $P(x)$ is a predicate and the domain of $x$ is the set of all real
numbers, let $R$ be "$\forall x \in \mathbb{Z}, P(x)$" let $S$ be
"$\forall x \in \mathbb{Q}, P(x)$", and let $T$ be
"$\forall x \in \mathbb{R}, P(x)$."
a. Find a definition for $P(x)$ (but do not use "$x \in \mathbb{Z}$") so that
$R$ is true and both $S$ and $T$ are false.
b. Find a definition for $P(x)$ (but do not use "$x \in \mathbb{Q}$") so that
both $R$ and $S$ are true and $T$ is false.
37. Consider the following sequence of digits: 0204. A person claims that all
the 1's in the sequence are to the left of all the 0's in the sequence. Is
this true? Justify your answer. (_Hint:_ Write the claim formally and write
a formal negation of it. Is the negation true or false?)
38. True or false? All occurrences of the letter _u_ in _Discrete Mathematics_
are lowercase. Justify your answer.
Rewrite each statement of 39-44 in if-then form.
39. Earning a grade of C- in this course is a sufficient condition for it to
count toward graduation.
40. Being divisible by 8 is a sufficient condition for being divisible by 4.
41. Being on time each day is a necessary condition for keeping this job.
42. Passing a comprehensive exam is a necessary condition for obtaining a
master's degree.
43. A number is prime only if it is greater than 1.
44. A polygon is square only if it has four sides.
Use the fact that the negation of a $\forall$ statement is a $\exists$ statement
and that the negation of an if-then statement is an _and_ statement to rewrite
each of the statements 45-48 without using the word _necessary_ or _sufficient_.
45. Being divisible by 8 is not a necessary condition for being divisible by 4.
46. Having a large income is not a necessary condition for a person to be happy.
47. Having a large income is not a sufficient condition for a person to be
happy.
48. Being a polynomial is not a sufficient condition for a function to have a
real root.
49. The computer scientists Richard Conway and David Gries once wrote:
> The absence of error messages during translation of a computer program is only
> a necessary and not a sufficient condition for reasonable [program]
> correctness.
Rewrite this statement without using the words _necessary_ or _sufficient_.
50. A frequent-flyer club brochure states, "You may select among carriers only
if they offer the same lowest fare." Assuming that "only if" has its formal,
logical meaning, does this statement guarantee that if two carriers offer
the same lowest fare, the customer will be free to choose between them?
Explain.

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chapter_3/notes.md
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@ -58,3 +58,84 @@ Let $P(x)$ and $Q(x)$ be predicates and suppose the domain of $x$ is $D$.
- The notation $P(x) \Leftrightarrow Q(x)$ means that $P(x)$ and $Q(x)$ have
identical truth sets, or, equivalently,
$\forall x, P(x) \leftrightarrow Q(x)$.
---
Page 145
**Theorem 3.2.1 Negation of a Universal Statement**
The negation of a statement of the form
$$ \forall \text{ in } D, Q(x) $$
is logically equivalent to a statement of the form
$$ \exists \text{ in } D \text{ such that } \neg Q(x) $$
Symbolically,
$$ \neg(\forall x \in D, Q(x)) \equiv \exists x \in D \text{ such that } \neg Q(x) $$
---
Page 146
**Theorem 3.2.2 Negation of an Existential Statement**
The negation of a statement of the form
$$ \exists \text{ in } D \text{ such that } Q(x) $$
is logically equivalent to a statement of the form
$$ \forall x \text{ in } D, \neg Q(x) $$
Symbolically,
$$ \neg(\exists x \in D \text{ such that } Q(x)) \equiv \forall x \in D, \neg Q(x) $$
---
Page 148
**Negation of a Universal Conditional Statement**
$$ \neg(\forall x, \text{ if } P(x) \text{ then } Q(x)) \equiv \exists x \text{ such that } P(x) \text{ and } \neg Q(x) $$
$$ \neg(\forall x, P(x) \to Q(x)) \equiv \exists x, (P(x) \wedge \neg Q(x)) $$
---
Page 150
**Definition**
Consider a statement of the form
$\forall x \in D, \text{ if } P(x) \text{ then } Q(x)$.
1. Its **contrapositive** is the statement
$\forall x \in D, \text{ if } \neg Q(x) \text{ then } \neg P(x)$.
2. Its **converse** is the statement
$\forall x \in D, \text{ if } Q(x) \text{ then } P(x)$.
3. Its **inverse** is the statement
$\forall x \in D, \text{ if } \neg P(x) \text{ then } \neg Q(x)$.
---
Page 151
**Definition**
- "$\forall x, r(x)$ is a **sufficient condition** for $s(x)$" means
"$\forall x, \text{ if } r(x) \text{ then } s(x)$."
- "$\forall x, r(x)$ is a **necessary condition** for $s(x)$" means
"$\forall x, \text{ if } \neg r(x) \text{ then } \neg s(x)$" or, equivalently,
"$\forall x, \text{ if } s(x) \text{ then } r(x)$."
- "$\forall x, r(x)$ **only if** $s(x)$" means
"$\forall x, \text{ if } \neg s(x) \text{ then } \neg r(x)$" or, equivalently,
"$\forall x, \text{ if } r(x) \text{ then } s(x)$."

38
chapter_3/test_yourself.md
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@ -25,3 +25,41 @@ true; every $x$ in $D$.
only if, $Q(x)$ is _______ for _______.
true; at least one $x$ in $D$.
---
**Test Yourself**
Page 152
1. A negation for "All $R$ have property $S$" is "There is _______ $R$ that
_______."
exists at least one; does not have property $S$.
2. A negation for "Some $R$ have property $S$" is "_______."
"No $R$ have property $S$."
3. A negation for "For every $x$, if $x$ has property $P$ then $x$ has property
$Q$" is "_______."
"There exists at least one $x$ such that $x$ has property $P$ and $x$ does not
have property $Q$."
4. The converse of "For every $x$, if $x$ has property $P$ then $x$ has property
$Q$" is "_______."
"For every $x$, if $x$ has property $Q$ then $x$ has property $P$."
5. The contrapositive of "For every $x$, if $x$ has property $P$ then $x$ has
property $Q$" is "_______."
"For every $x$, if $x$ does not have property $Q$, then $x$ does not have
property $P$."
6. The inverse of "For every $x$, if $x$ has property $P$ then $x$ has property
$Q$" is "_______."
"For every $x$, if $x$ does not have property $P$, then $x$ does not have
property $Q$."