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**Exercise Set 2.1**
Page 74
In each of 1-4 represent the common form of each argument using letters to stand
for component sentences, and fill in the blanks so that the argument in part (b)
has the same logical form as the argument in part (a).
1.
a.
If all integers are rational, then the number $1$ is rational.
All integers are rational.
Therefore, the number $1$ is rational.
b.
If all algebraic expressions can be written in prefix notation ,then
______.
______.
Therefore, $(a + 2b)(a^2 - b)$ can be written in prefix notation.
2.
a.
If all computer programs contain errors, then this program contains an error.
This program does not contain an error.
Therefore, it is not the case that all computer programs contain errors.
b.
If ______, then ______.
2 is not odd.
Therefore, it is not the case that all prime numbers are odd.
3.
a.
This number is even or this number is odd.
This number is not even.
Therefore, this number is odd.
b.
______ or logic is confusing.
My mind is not shot.
Therefore, ______.
4.
a.
If the program syntax is faulty, then the computer will generate an error
message.
If the computer generates an error message, then the program will not run.
Therefore, if the program syntax is faulty, then the program will not run.
b.
If this simple graph ______, then it is complete.
If this graph ______, then any two of its vertices can be joined by a path.
Therefore, if this simple graph has 4 vertices and 6 edges, then ______.
5. Indicate which of the following sentences are statements.
a. 1,024 is the smallest four-digit number that is a perfect square.
b. She is a mathematics major.
c. $128 = 2^6$
d. $x = 2^6$
Write the statements in 6-9 in symbolic form using the symbols $\neg$, $\wedge$,
$\vee$ and the indicated letters to represent component statements.
6. Let $s = $ "stocks are increasing" and $i = $ "interest rates are steady."
a. Stocks are increasing but interest rates are steady.
b. Neither are stocks increasing nor are interest rates steady.
7. Juan is a math major but not a computer science major. ($m = $ "Juan is a
math major," $c = $ "Juan is a computer science major")
8. Let $h = $ "John is healthy," $w = $ "John is wealthy," and $s = $ "John is
wise."
a. John is healthy and wealthy but not wise.
b. John is not wealthy but he is healthy and wise.
c. John is neither healthy, wealthy, nor wise.
d. John is neither wealthy nor wise, but he is healthy.
e. John is wealthy, but he is not both healthy and wise.
9. Let $p = $ "$x > 5$," $q = $ "$x = 5$," and $r = $ "$10 > x$."
a. $x \geq 5$
b. $10 > x > 5$
c. $10 > x \geq 5$
10. Let $p$ be the statement "DATAENDFLAG is off," $q$ the statement "ERROR
equals 0," and $r$ the statement "SUM is less than 1,000." Express the
following sentences in symbolic notation.
a. DATAENDFLAG is off, ERROR equals 0, and SUM is less than 1,000.
b. DATAENDFLAG is off but ERROR is not equal to 0.
c. DATAENDFLAG is off; however, ERROR is not 0 or SUM is greater than or equal
to 1,000.
e. Either DATAENDFLAG is on or it is the case that both ERROR equals 0 and SUM
is less than 1,000.
11. In the following sentence, is the word _or_ used in its inclusive or
exclusive sense? A team wins the playoffs if it wins two games in a row or a
total of three games.
Write truth tables for the statement forms 12-15.
12. $\neg p \wedge q$
13. $\neg (p \wedge q) \vee (p \vee q)$
14. $p \wedge (q \wedge r)$
15. $p \wedge (\neg q \vee r)$
Determine whether the statement forms in 16-24 are logically equivalent. In each
case, construct a truth table and include a sentence justifying your answer.
Your sentence should show that you understand the meaning of logical
equivalence.
16. $p \vee (p \wedge q) \text{ and } p$
17. $\neg (p \wedge q) \text{ and } \neg p \wedge \neg q$
18. $p \vee \mathbf{t} \text{ and } \mathbf{t}$
19. $p \wedge \mathbf{t} \text{ and } $
20. $p \wedge \mathbf{c} \text{ and } p \vee \mathbf{c}$
21. $(p \wedge q) \wedge r \text{ and } p \wedge (q \wedge r)$
22. $p \wedge (q \vee r) \text{ and } (p \wedge q) \vee (p \wedge r)$
23. $(p \wedge q) \vee r \text{ and } p \wedge (q \vee r)$
24. $(p \vee q) \vee (p \wedge r) \text{ and } (p \vee q) \wedge r$
Use De Morgan's laws to write negations for the statements in 25-30.
25. Hal is a math major and Hal's sister is a computer science major.
26. Sam is an orange belt and Kate is a red belt.
27. The connector is loose or the machine is unplugged.
28. The train is late or my watch is fast.
29. This computer program has a logical error in the first ten lines or it is
being run with an incomplete data set.
30. The dollar is at an all-time high and the stock market is at a record low.
31. Let $s$ be a string of length 2 with characters from $\{0, 1, 2\}$, and
define statements $a$, $b$, $c$, and $d$ as follows:
$a = $ "the first character of $s$ is 0"
$b = $ "the first character of $s$ is 1"
$c = $ "the second character of $s$ is 1"
$c = $ "the second character of $s$ is 2".
Describe the set of all strings for which each of the following is true.
a. $(a \vee b) \wedge (c \vee d)$
b. $(\neg(a \vee b)) \wedge (c \vee d)$
c. $((\neg a) \vee b) \wedge (c \vee (\neg d))$
Assume $x$ is a particular real number and use De Morgan's laws to write
negations for the statements in 32-37.
32. $-2 < x < 7$
33. $-10 < x < 2$
34. $x < 2 \text{ or } x > 5$
35. $x \leq -1 \text{ or } x > 1$
36. $1 > x \geq -3$
37. $0 > x \geq -7$
In 38 and 39, imagine that _num_orders_ and _num_instock_ are particular values,
such as might occur during execution of a computer program. Write negations for
the following statements.
38. $(\text{num_orders } > 100 \text{ and } \text{num_instock } \leq 500) \text{ or } \text{num_instock } < 200$
39. $(\text{num_orders } < 50 \text{ and } \text{num_instock } > 300) \text{ or } (50 \leq \text{ num_orders } < 75 \text{ and } \text{num_instock} > 500)$
Use truth tables to establish which of the statement forms in 40-43 are
tautologies and which are contradictions.
40. $(p \wedge q) \vee (\neg p \vee (p \wedge \neg q))$
41. $(p \wedge \neg q) \wedge (\neg p \vee q)$
42. $((\neg p \wedge q) \wedge (q \wedge r)) \wedge \neg q$
43. $(\neg p \vee q) \vee (p \wedge \neg q)$
44. Recall that $a < x < b$ means that $a < x$ and $x < b$. Also $a \leq b$
means that $a < b$ or $a = b$. Find all real numbers that satisfy the
following inequalities.
a. $2 < x \leq 0$
b. $1 \leq x < -1$
45. Determine whether the statements in (a) and (b) are logically equivalent.
a. Bob is both a math and computer science major and Ann is a math major, but
Ann is not both a math and computer science major.
b. It is not the case that both Bob and Ann are both math and computer science
majors, but it is the case that Ann is a math major and Bob is both a math and
computer science major.
46. Let the symbol $\oplus$ denote _exclusive or_; so
$p \plus q \equiv (p \vee q) \wedge \neg(p \wedge q)$. Hence the truth table
for $p \plus q$ is as follows:
| $p$ | $q$ | $p \plus q$ |
| --- | --- | ----------- |
| T | T | F |
| T | F | T |
| F | T | T |
| F | F | F |
a. Find simpler statement forms that are logically equivalent to $p \plus p$ and
$(p \oplus p) \oplus p$.
b. Is $(p \oplus q) \oplus r \equiv p \oplus (q \oplus r)$? Justify your answer.
c. Is $(p \oplus q) \wedge r \equiv (p \wedge r) \oplus (q \wedge r)$? Justify
your answer.
47. In logic and in standard English, a double negative is equivalent to a
positive. There is one fairly common English usage in which a "double
positive" is equivalent to a negative. What is it? Can you think of others?
In 48 and 49 below, a logical equivalence is derived from Theorem 2.1.1. Supply
a reason for each step.
48.
$$ p \vee \neg q \vee (p \wedge q) \equiv p \wedge (\neg q \vee q) \text{ by (a)} $$
$$ \quad \equiv p \wedge (q \vee \neg q) \text{ by (b)} $$
$$ \quad \equiv p \wedge \mathbf{t} \text{ by (c)} $$
$$ \quad \equiv p \text{ by (d)} $$
Therefore, $(p \wedge \neg q) \vee (p \wedge q) \equiv p$.
49.
$$ (p \vee \neg q) \wedge (\neg p \vee \neg q) $$
$$ \quad \equiv (\neg q \vee p) \wedge (\neg q \vee \neg p) \text{ by (a)} $$
$$ \quad \equiv \neg q \vee (p \wedge \neg p) \text{ by (b)} $$
$$ \quad \equiv q \vee \mathbf{c} \text{ by (c)} $$
$$ \quad \equiv \neg q \text{ by (d)} $$
Therefore, $(p \vee \neg q) \wedge (\neg p \vee \neg q) \equiv \neg q$.
Use Theorem 2.1.1 to verify the logical equivalences in 50-54. Supply a reason
for each step.
50. $(p \wedge \neg q) \vee p \equiv p$
51. $p \wedge (\neg q \vee p) \equiv p$
52. $\neg(p \vee \neg q) \vee (\neg p \wedge \neg q) \equiv \neg p$
53. $\neg((\neg p \wedge q) \vee (\neg p \wedge \neg q)) \vee (p \wedge q) \equiv p$
54. $(p \wedge (\neg(\neg p \vee q))) \vee (p \wedge q) \equiv p$

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Page 61
**Definition**
A **statement** (or **proposition**) is a sentence that is true or false but not
both.
---
Page 63
**Definition**
If $p$ is a statement variable, the **negation** of $p$ is "not $p$" or "It is
not the case that $p$" and is denoted $\neg p$. It has opposite truth value from
$p$: if $p$ is true, $\neg p$ is false; if $p$ is false, $\neg p$ is true.
---
Page 64
**Definition**
If $p$ and $q$ are statement variables, the **conjunction** of $p$ and $q$ is
"$p$ and $q$", denoted $p \wedge q$. It is true when, and only when, both $p$
and $q$ are true. If either $p$ or $q$ is false, or if both are false,
$p \wedge q$ is false.
---
Page 64
**Definition**
If $p$ and $q$ are statement variables, the **disjunction** of $p$ and $q$ is
"$p$ or $q$", denoted $p \vee q$. It is true when either $p$ is true, or $q$ is
true, or both $p$ and $q$ are true; it is false only when both $p$ and $q$ are
false.
---
Page 65
**Definition**
A **statement form** (or **propositional form**) is an expression made up of
statement variables (such as $p$, $q$, and $r$) and logical connectives (such as
$\neg$, $\wedge$, and $\vee$) that becomes a statement when actual statements
are substituted for the component statement variables. The **truth table** for a
given statement form displays the truth values that correspond to all possible
combinations of truth values for its component statement variables.
---
Page 67
**Definition**
Two _statement forms_ are called **logically equivalent** if, and only if, they
have identical truth values for each possible substitution of statements for
their statement variables. The logical equivalence of statements forms $P$ and
$Q$ is denoted by writing $P \equiv Q$.
Two _statements_ are called **logically equivalent** if, and only if, they have
logically equivalent forms when identical component statement variables are used
to replace identical component statements.
---
Page 69
**De Morgan's Laws**
The negation of an _and_ statement is logically equivalent to the _or_ statement
in which each component is negated.
The negation of an _or_ statement is logically equivalent to the _and_ statement
in which each component is negated.
---
Page 71
**Definition**
A **tautology** is a statement form that is always true regardless of the truth
values of the individual statements substituted for its statement variables. A
statement whose form is a tautology is a **tautological statement**.
A **contradiction** is a statement form that is always false regardless of the
truth values of the individual statements substituted for its statement
variables. A statement whose form is a contradiction is a **contradictory
statement**.

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**Test Yourself**
Page 73
1. An _and_ statement is true when, and only when, both components are _______.
**Solution**
True.
2. An _or_ statement is false when, and only when, both components are _______.
**Solution**
False.
3. Two statement forms are logically equivalent when, and only when, they always
have _______.
**Solution**
The same truth values.
4. De Morgan's laws says (1) that the negation of an _and_ statement is
logically equivalent to the _______ statement in which each component is
_______, and (2) that the negation of an _or_ statement is logically
equivalent to the _______ statement in which each component is _______.
**Solution**
or; negated; and; negated.
5. A tautology is a statement that is always _______.
**Solution**
true
6. A contradiction is a statement that is always _______.
**Solution**
false