Logic_Exercises_2
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Logic Exercises - Part 2
Rewrite the statements in if-then form.
1. This loop will repeat exactly
N
times if it does not contain a
stop
or a
go to
.
2. I am on time for work if I catch the 8:05 bus.
Solution:
If I catch the 8:05 bus then I am on time for work.
3. Freeze or I’ll shoot.
4. Fix my ceiling or I won’t pay my rent.
Construct truth tables for the statement forms.
5.
∼
p
∨
q
→∼
q
6. (
p
∨
q
)
∨
(
∼
p
∧
q
)
→
q
7.
p
∧ ∼
q
→
r
8.
∼
p
∨
q
→
r
9.
p
∧ ∼
r
↔
q
∨
r
Solution:
p
q
r
∼
r
p
∧ ∼
r
q
∨
r
p
∧ ∼
r
↔
q
∨
r
T
T
T
F
F
T
F
T
T
F
T
T
T
T
T
F
T
F
F
T
F
T
F
F
T
T
F
F
F
T
T
F
F
T
F
F
T
F
T
F
T
F
F
F
T
F
F
T
F
F
F
F
T
F
F
T
10. (
p
→
r
)
↔
(
q
→
r
)
11. (
p
→
(
q
→
r
))
↔
((
p
∧
q
)
→
r
)
1
Solution:
p
q
r
q
→
r
p
→
(
q
→
r
)
p
∧
q
(
p
∧
q
)
→
r
(
p
→
(
q
→
r
))
↔
((
p
∧
q
)
→
r
)
T
T
T
T
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
T
T
F
T
T
T
F
F
T
T
F
T
T
F
T
T
T
T
F
T
T
F
T
F
F
T
F
T
T
F
F
T
T
T
F
T
T
F
F
F
T
T
F
T
T
12. Use the logical equivalence
p
∨
q
→
r
≡
(
p
→
r
)
∧
(
q
→
r
), to rewrite the
following statement. (Assume that
x
represents a fixed real number.)
If
x >
2 or
x <
−
2, then
x
2
>
4.
Solution:
(
x >
2
→
x
2
>
4
)
∧
(
x <
−
2
→
x
2
>
4
)
13. Use truth tables to verify the following logical equivalences. Include a few
words of explanation with your answers.
(a)
p
→
q
≡∼
p
∨
q
(b)
∼
(
p
→
q
)
≡
p
∧ ∼
q
14.
a.
Show that the following statement forms are all logically equivalent.
p
→
q
∨
r
,
p
∧ ∼
q
→
r
, and
p
∧ ∼
r
→
q
b.
Use the logical equivalences established in part (a) to rewrite the
following sentence in two different ways.
(Assume that
n
represents a
fixed integer.)
If
n
is prime, then
n
is odd or
n
is 2.
Solution:
(a)
p
→
q
∨
r
≡∼
p
∨
(
q
∨
r
)
(Using
p
→
q
≡∼
p
∨
q
)
p
∧ ∼
q
→
r
≡∼
(
p
∧ ∼
q
)
∨
r
(Using
p
→
q
≡∼
p
∨
q
)
≡
(
∼
p
∨
q
)
∨
r
(De Morgan’s Law)
≡∼
p
∨
(
q
∨
r
)
(Associative Law)
p
∧ ∼
r
→
q
≡∼
(
p
∧ ∼
r
)
∨
q
(Using
p
→
q
≡∼
p
∨
q
)
≡
(
∼
p
∨
r
)
∨
q
(De Morgan’s Law)
≡∼
p
∨
(
r
∨
q
)
(Associative Law)
2
From the above, it’s clear that all three statements are equivalent to
the form
∼
p
∨
(
q
∨
r
).
(b)
Solution:
Using the equivalences, the statement can be rewritten
as:
i.
n
is not prime or,
n
is odd or
n
is 2.
ii. If
n
is prime and
n
is not odd, then
n
is 2.
iii. If
n
is prime and
n
is not 2, then
n
is odd.
15. Determine whether the following statement forms are logically equivalent:
p
→
(
q
→
r
) and (
p
→
q
)
→
r
Write each of the two statements in symbolic form and determine
whether they are logically equivalent. Include a truth table and
a few words of explanation.
16. If you paid full price, you didn’t buy it at Crown Books. You didn’t buy
it at Crown Books or you paid full price.
17. If 2 is a factor of
n
and 3 is a factor of
n
, then 6 is a factor of
n
. 2 is not
a factor of
n
or 3 is not a factor of
n
or 6 is a factor of
n
.
18. Write each of the following three statements in symbolic form and deter-
mine which pairs are logically equivalent. Include truth tables and a few
words of explanation.
(a) If it walks like a duck and it talks like a duck, then it is a duck.
(b) Either it does not walk like a duck or it does not talk like a duck, or
it is a duck.
(c) If it does not walk like a duck and it does not talk like a duck, then
it is not a duck.
Solution:
Let:
p
: It walks like a duck.
q
: It talks like a duck.
r
: It is a duck.
p
q
r
p
∧
q
→
r
∼
p
∨ ∼
q
∨
r
∼
p
∧ ∼
q
→∼
r
T
T
T
T
T
T
T
T
F
F
F
T
T
F
T
T
T
T
T
F
F
T
T
T
F
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
T
T
F
F
F
T
T
F
The first and second statements are logically equivalent.
However, the
third statement specifies that lacking both duck-like characteristics di-
rectly implies it’s not a duck.
3
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19. True or false?
The negation of “If Sue is Luiz’s mother, then Ali is his
cousin” is “If Sue is Luiz’s mother, then Ali is not his cousin.”
20. Write negations for each of the following statements.
(Assume that all
variables represent fixed quantities or entities, as appropriate.)
(a) If
P
is a square, then
P
is a rectangle.
(b) If today is New Year’s Eve, then tomorrow is January.
(c) If the decimal expansion of
r
is terminating, then
r
is rational.
(d) If
n
is prime, then
n
is odd or
n
is 2.
(e) If
x
is nonnegative, then
x
is positive or
x
is 0.
(f) If Tom is Ann’s father, then Jim is her uncle and Sue is her aunt.
(g) If
n
is divisible by 6, then
n
is divisible by 2 and
n
is divisible by 3.
Solution:
To negate a conditional statement ”If P then Q”, we can use the form
”P and not Q”.
(a)
P
is a square and
P
is not a rectangle.
(b) Today is New Year’s Eve and tomorrow is not January.
(c) The decimal expansion of
r
is terminating and
r
is not rational.
(d)
n
is prime and
n
is not odd and
n
is not 2.
(e)
x
is nonnegative and
x
is not positive and
x
is not 0.
(f) Tom is Ann’s father and (Jim is not her uncle or Sue is not her aunt).
(g)
n
is divisible by 6 and (
n
is not divisible by 2 or
n
is not divisible by
3).
21. Suppose that
p
and
q
are statements so that
p
→
q
is false. Find the truth
values of each of the following:
(a)
∼
p
→
q
(b)
p
∨
q
(c)
q
→
p
22. Write contrapositives for the statements of exercise 20.
23. Write the converse and inverse for each statement of exercise 20.
Solution:
(a) Converse: If
P
is a rectangle, then
P
is a square.
Inverse: If
P
is not a square, then
P
is not a rectangle.
(b) Converse: If tomorrow is January, then today is New Year’s Eve.
Inverse: If today is not New Year’s Eve, then tomorrow is not Jan-
uary.
4
(c) Converse: If
r
is rational, then the decimal expansion of
r
is termi-
nating.
Inverse: If the decimal expansion of
r
is not terminating, then
r
is
not rational.
(d) Converse: If
n
is odd or
n
is 2, then
n
is prime.
Inverse: If
n
is not prime, then
n
is not odd and
n
is not 2.
(e) Converse: If
x
is positive or
x
is 0, then
x
is nonnegative.
Inverse: If
x
is negative, then
x
is not positive and
x
is not 0.
(f) Converse: If Jim is Ann’s uncle and Sue is her aunt, then Tom is
Ann’s father.
Inverse: If Tom is not Ann’s father, then Jim is not her uncle or Sue
is not her aunt.
(g) Converse:
If
n
is divisible by 2 and
n
is divisible by 3, then
n
is
divisible by 6.
Inverse: If
n
is not divisible by 6, then
n
is not divisible by 2 or
n
is
not divisible by 3.
Use truth tables to establish the truth of each statement.
24. A conditional statement is not logically equivalent to its converse.
25. A conditional statement is not logically equivalent to its inverse.
26. A conditional statement and its contrapositive are logically equivalent to
each other.
Solution:
P
Q
P
→
Q
¬
Q
¬
P
¬
Q
→ ¬
P
T
T
T
F
F
T
T
F
F
T
F
F
F
T
T
F
T
T
F
F
T
T
T
T
27. The converse and inverse of a conditional statement are logically equivalent
to each other.
Solution:
P
Q
Q
→
P
¬
P
¬
Q
¬
P
→ ¬
Q
T
T
T
F
F
T
T
F
T
F
T
T
F
T
F
T
F
F
F
F
T
T
T
T
If statement forms
P
and
Q
are logically equivalent, then
P
↔
Q
is a tautology. Conversely, if
P
↔
Q
is a tautology, then
P
and
Q
are logically equivalent. Use
↔
to convert each of the logical
equivalences to a tautology.
Then use a truth table to verify
each tautology.
5
28.
p
→
(
q
∨
r
)
≡
(
p
∧ ∼
q
)
→
r
Solution:
p
q
r
q
∨
r
p
→
(
q
∨
r
)
¬
q
p
∧ ¬
q
(
p
∧ ¬
q
)
→
r
p
→
(
q
∨
r
)
↔
(
p
∧ ¬
q
)
→
r
T
T
T
T
T
F
F
T
T
T
T
F
T
T
F
F
T
T
T
F
T
T
T
T
T
T
T
T
F
F
F
F
T
T
F
T
F
T
T
T
T
F
F
T
T
F
T
F
T
T
F
F
T
T
F
F
T
T
T
T
F
T
T
F
F
F
F
T
T
F
T
T
29.
p
∧
(
q
∨
r
)
≡
(
p
∧
q
)
∨
(
p
∧
r
)
30.
p
→
(
q
→
r
)
≡
(
p
∧
q
)
→
r
Rewrite each of the statements as a conjunction of two if-then
statements.
31. This quadratic equation has two distinct real roots if, and only if, its
discriminant is greater than zero.
32. This integer is even if, and only if, it equals twice some integer.
Solution:
If this integer is even, then it equals twice some integer and If this integer
equals twice some integer, then it is even.
Rewrite the statements in if-then form in two ways, one of which
is the contrapositive of the other.
33. The Cubs will win the pennant only if they win tomorrow’s game.
34. Sam will be allowed on Signe’s racing boat only if he is an expert sailor.
Solution:
•
If Sam will be allowed on Signe’s racing boat, then he is an expert
sailor.
•
If Sam is not an expert sailor, then he will not be allowed on Signe’s
racing boat.
35. Taking the long view on your education, you go to the Prestige Corpora-
tion and ask what you should do in college to be hired when you graduate.
The personnel director replies that you will be hired only if you major in
mathematics or computer science, get a B average or better, and take
accounting.
You do, in fact, become a math major, get a B+ average,
and take accounting. You return to Prestige Corporation, make a formal
application, and are turned down. Did the personnel director lie to you?
6
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Rewrite the statements in if-then form.
36. Catching the 8:05 bus is a sufficient condition for my being on time for
work.
37. Having two 45
◦
angles is a sufficient condition for this triangle to be a
right triangle.
Solution:
•
If this triangle has two 45
◦
angles, then it is a right triangle.
Use the contrapositive to rewrite the statements in if-then form
in two ways.
38. Being divisible by 3 is a necessary condition for this number to be divisible
by 9.
39. Doing homework regularly is a necessary condition for Jim to pass the
course.
Solution:
•
If Jim does not do homework regularly, then he does not pass the
course.
•
If Jim passes the course, then he does homework regularly.
Note that “a sufficient condition for
s
is
r
” means
r
is a sufficient
condition for
s
and that “a necessary condition for
s
is
r
” means
r
is a necessary condition for
s
. Rewrite the statements in if-then
form.
40. A sufficient condition for Jon’s team to win the championship is that it
win the rest of its games.
41. A necessary condition for this computer program to be correct is that it
not produce error messages during translation.
Solution:
•
If the computer program is correct, then it does not produce error
messages during translation.
42. “If compound X is boiling, then its temperature must be at least 150
◦
C
.”
Assuming that this statement is true, which of the following must also be
true?
(a) If the temperature of compound X is at least 150
◦
C
, then compound
X is boiling.
(b) If the temperature of compound X is less than 150
◦
C
, then compound
X is not boiling.
7
(c) Compound X will boil only if its temperature is at least 150
◦
C
.
(d) If compound X is not boiling, then its temperature is less than 150
◦
C
.
(e) A necessary condition for compound X to boil is that its temperature
be at least 150
◦
C
.
(f) A sufficient condition for compound X to boil is that its temperature
be at least 150
◦
C
.
In the following exercises: (a) use the logical equivalences
p
→
q
≡∼
p
∨
q
and
p
↔
q
≡
(
∼
p
∨
q
)
∧
(
∼
q
∨
p
)
to rewrite the given
statement forms without using the symbol
→
or
↔
, and (b)
use the logical equivalence
p
∨
q
≡∼
(
∼
p
∧ ∼
q
)
to rewrite each
statement form using only
∧
and
∼
.
43.
p
∧ ∼
q
→
r
44.
p
∨ ∼
q
→
r
∨
q
Solution:
(a)
∼
(
p
∨ ∼
q
)
∨
(
r
∨
q
)
(b)
∼
(
∼
(
∼
p
∧
q
)
∧ ∼
r
∧ ∼
q
)
45. (
p
→
r
)
↔
(
q
→
r
)
46. (
p
→
(
q
→
r
))
↔
((
p
∧
q
)
→
r
)
47. Given any statement form, is it possible to find a logically equivalent form
that uses only
∼
and
∧
? Justify your answer.
Use truth tables to determine whether the argument forms are
valid. Indicate which columns represent the premises and which
represent the conclusion, and include a sentence explaining how
the truth table supports your answer. Your explanation should
show that you understand what it means for a form of argument
to be valid or invalid.
48.
p
→
q
q
→
p
∴
p
∨
q
49.
p
p
→
q
¬
q
∨
r
∴
r
50.
p
∨
q
p
→ ¬
q
p
→
r
∴
r
8
Solution:
p
q
r
¬
q
p
∨
q
p
→ ¬
q
p
→
r
r
T
T
T
F
T
F
T
T
T
T
F
F
T
F
F
F
T
F
T
T
T
T
T
T
T
F
F
T
T
T
F
F
F
T
T
F
T
T
T
T
F
T
F
F
T
T
T
F
F
F
T
T
F
T
T
T
F
F
F
T
F
T
T
F
The critical rows with true premises are rows 3, 5, and 6. On row 6, the
conclusion is false; therefore, the argument is invalid.
51. (
p
∧
q
)
→ ¬
r
p
∨ ¬
q
¬
q
→
p
∴
¬
r
52.
p
→
r
q
→
r
∴
(
p
∨
q
)
→
r
53.
p
→
(
q
∨
r
)
¬
q
∨ ¬
r
∴
¬
p
∨ ¬
r
Use truth tables to show that the following forms of argument
are invalid.
54.
p
→
q
q
∴
p
(converse error)
55.
p
→
q
∼
p
∴
∼
q
(inverse error)
Use symbols to write the logical form of each argument, and
then use a truth table to test the argument for validity. Indicate
which columns represent the premises and which represent the
conclusion, and include a few words of explanation showing that
you understand the meaning of validity.
56. If Tom is not on team
A
, then Hua is on team
B
.
If Hua is not on team
B
, then Tom is on team
A
.
∴
Tom is not on team
A
or Hua is not on team
B
.
9
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Solution:
Let’s represent the given statements using logical symbols:
•
p
: Tom is on team
A
•
q
: Hua is on team
B
The given statements can be represented as:
(a)
¬
p
→
q
(If Tom is not on team
A
, then Hua is on team
B
)
(b)
¬
q
→
p
(If Hua is not on team
B
, then Tom is on team
A
)
(c) Conclusion:
¬
p
∨ ¬
q
(Tom is not on team
A
or Hua is not on team
B
)
Now, let’s create a truth table for these statements:
p
q
¬
p
¬
q
¬
p
→
q
¬
q
→
p
¬
p
∨ ¬
q
T
T
F
F
T
T
F
T
F
F
T
T
T
T
F
T
T
F
T
T
T
F
F
T
T
F
F
T
From the truth table, we can analyze the validity of the argument:
The columns representing the premises are:
•
¬
p
→
q
•
¬
q
→
p
The column representing the conclusion is:
•
¬
p
∨ ¬
q
In row 1, both premises are true, but the conclusion is false, making the
argument invalid.
57. Oleg is a math major or Oleg is an economics major.
If Oleg is a math major, then Oleg is required to take Math 362.
∴
Oleg is an economics major or Oleg is not required to take Math 362.
Some of the arguments are valid, whereas others exhibit the
converse or the inverse error. Use symbols to write the logical
form of each argument.
If the argument is valid, identify the
rule of inference that guarantees its validity.
Otherwise, state
whether the converse or the inverse error is made.
58. If Jules solved this problem correctly, then Jules obtained the answer 2.
Jules obtained the answer 2.
∴
Jules solved this problem correctly.
10
59. This real number is rational or it is irrational.
This real number is not rational.
∴
This real number is irrational.
Solution:
Let’s represent the given statements using logical symbols:
•
p
: This real number is rational.
•
q
: This real number is irrational.
The given statements can be represented as:
(a)
p
∨
q
(This real number is rational or it is irrational.)
(b)
¬
p
(This real number is not rational.)
(c) Conclusion:
q
(This real number is irrational.)
This is a valid argument and follows the rule of Disjunctive Syllogism (or
Elimination). If one part of an ”or” statement is false, the other must be
true.
60. If I go to the movies, I won’t finish my homework.
If I don’t finish my
homework, I won’t do well on the exam tomorrow.
∴
If I go to the movies, I won’t do well on the exam tomorrow.
61. If this number is larger than 2, then its square is larger than 4.
This number is not larger than 2.
∴
The square of this number is not larger than 4.
Solution:
Let’s represent the given statements using logical symbols:
•
p
: This number is larger than 2.
•
q
: The square of this number is larger than 4.
The given statements can be represented as:
(a)
p
→
q
(If this number is larger than 2, then its square is larger than
4.)
(b)
¬
p
(This number is not larger than 2.)
(c) Conclusion:
¬
q
(The square of this number is not larger than 4.)
The form of the argument is Denying the Antecedent or the Inverse Error,
which is a formal fallacy. This means that even if the premises are true,
the conclusion does not necessarily follow. In other words, just because a
number is not larger than 2 doesn’t guarantee that its square isn’t larger
than 4. (example: (
−
3)
2
= 9)
11
62. If there are as many rational numbers as there are irrational numbers,
then the set of all irrational numbers is infinite.
The set of all irrational numbers is infinite.
∴
There are as many rational numbers as there are irrational numbers.
63. If at least one of these two numbers is divisible by 6, then the product of
these two numbers is divisible by 6.
Neither of these two numbers is divisible by 6.
∴
The product of these two numbers is not divisible by 6.
64. If this computer program is correct, then it produces the correct output
when run with the test data my teacher gave me.
This computer program produces the correct output when run with the
test data my teacher gave me.
∴
This computer program is correct.
65. Sandra knows Java and Sandra knows C++.
∴
Sandra knows C++.
66. If I get a Christmas bonus, I’ll buy a stereo.
If I sell my motorcycle, I’ll buy a stereo.
∴
If I get a Christmas bonus or I sell my motorcycle, then I’ll buy a stereo.
67. Explain in your own words what distinguishes a valid form of argument
from an invalid one.
68. Given the following information about a computer program, find the mis-
take in the program.
(a) There is an undeclared variable or there is a syntax error in the first
five lines.
(b) If there is a syntax error in the first five lines, then there is a missing
semicolon or a variable name is misspelled.
(c) There is not a missing semicolon.
(d) There is not a misspelled variable name.
69. In the back of an old cupboard you discover a note signed by a pirate
famous for his bizarre sense of humor and love of logical puzzles. In the
note he wrote that he had hidden treasure somewhere on the property. He
listed five true statements (a–e below) and challenged the reader to use
them to figure out the location of the treasure.
(a) If this house is next to a lake, then the treasure is not in the kitchen.
(b) If the tree in the front yard is an elm, then the treasure is in the
kitchen.
(c) This house is next to a lake.
12
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(d) The tree in the front yard is an elm or the treasure is buried under
the flagpole.
(e) If the tree in the back yard is an oak, then the treasure is in the
garage.
Where is the treasure hidden?
70. You are visiting the island described before and have the following en-
counters with natives.
(a) Two natives A and B address you as follows:
A says: Both of us are knights.
B says: A is a knave.
What are A and B?
(b) Another two natives C and D approach you but only C speaks.
C says: Both of us are knaves.
What are C and D?
(c) You then encounter natives E and F.
E says: F is a knave.
F says: E is a knave.
How many knaves are there?
(d) Finally, you meet a group of six natives, U, V, W, X, Y, and Z, who
speak to you as follows:
U says: None of us is a knight.
V says: At least three of us are knights.
W says: At most three of us are knights.
X says: Exactly five of us are knights.
Y says: Exactly two of us are knights.
Z says: Exactly one of us is a knight.
Which are knights and which are knaves?
71. The famous detective Percule Hoirot was called in to solve a baffling mur-
der mystery. He determined the following facts:
(a) Lord Hazelton, the murdered man, was killed by a blow on the head
with a brass candlestick.
(b) Either Lady Hazelton or a maid, Sara, was in the dining room at the
time of the murder.
(c) If the cook was in the kitchen at the time of the murder, then the
butler killed Lord Hazelton with a fatal dose of strychnine.
(d) If Lady Hazelton was in the dining room at the time of the murder,
then the chauffeur killed Lord Hazelton.
(e) If the cook was not in the kitchen at the time of the murder, then
Sara was not in the dining room when the murder was committed.
13
(f) If Sara was in the dining room at the time the murder was committed,
then the wine steward killed Lord Hazelton.
Is it possible for the detective to deduce the identity of the murderer
from these facts?
If so, who did murder Lord Hazelton?
(Assume
there was only one cause of death.)
72. Sharky, a leader of the underworld, was killed by one of his own band
of four henchmen. Detective Sharp interviewed the men and determined
that all were lying except for one. He deduced who killed Sharky on the
basis of the following statements:
(a) Socko: Lefty killed Sharky.
(b) Fats: Muscles didn’t kill Sharky.
(c) Lefty:
Muscles was shooting craps with Socko when Sharky was
knocked off.
(d) Muscles: Lefty didn’t kill Sharky.
Who killed Sharky ?
In the following exercises a set of premises and a conclusion are
given. Use valid argument forms to deduce the conclusion from
the premises, giving a reason for each step. Assume all variables
are statement variables.
73.
(a)
∼
p
∨
q
→
r
(b)
s
∨ ∼
q
(c)
∼
t
(d)
p
→
t
(e)
∼
p
∧
r
→∼
s
(f)
∴
∼
q
74.
(a)
p
∨
q
(b)
q
→
r
(c)
p
∧
s
→
t
(d)
∼
r
(e)
∼
q
→
u
∧
s
(f)
∴
t
75.
(a)
∼
p
→
r
∧ ∼
s
(b)
t
→
s
(c)
u
→∼
p
(d)
∼
w
(e)
u
∨
w
14
(f)
∴
∼
t
76.
(a)
p
→
q
(b)
r
∨
s
(c)
∼
s
→∼
t
(d)
∼
q
∨
s
(e)
∼
s
(f)
∼
p
∧
r
→
u
(g)
w
∨
t
(h)
∴
u
∧
w
77. Let
P
(
x
) be the predicate “
x >
1
x
”.
(a) Write
P
(2),
P
(
1
2
)
,
P
(
−
1),
P
(
−
1
2
)
, and
P
(
−
8), and indicate which
of these statements are true and which are false.
(b) Find the truth set of
P
(
x
) if the domain of
x
is
R
, the set of all real
numbers.
(c) If the domain is the set
R
+
of all positive real numbers, what is the
truth set of
P
(
x
)?
78. Let
Q
(
n
) be the predicate “
n
2
≤
30”.
(a) Write
Q
(2),
Q
(
−
2),
Q
(7), and
Q
(
−
7), and indicate which of these
statements are true and which are false.
(b) Find the truth set of
Q
(
n
) if the domain of
n
is
Z
, the set of all
integers.
(c) If the domain is the set
Z
+
of all positive integers, what is the truth
set of
Q
(
n
)?
79. Let
Q
(
x, y
) be the predicate “If
x < y
then
x
2
< y
2
” with domain for
both
x
and
y
being the set
R
of real numbers.
(a) Explain why
Q
(
x, y
) is false if
x
=
−
2 and
y
= 1.
(b) Give values different from those in part (a) for which
Q
(
x, y
) is false.
(c) Explain why
Q
(
x, y
) is true if
x
= 3 and
y
= 8.
(d) Give values different from those in part (c) for which
Q
(
x, y
) is true.
80. Let
R
(
m, n
) be the predicate “If
m
is a factor of
n
2
then
m
is a factor of
n
,” with domain for both
m
and
n
being the set
Z
of integers.
(a) Explain why
R
(
m, n
) is false if
m
= 25 and
n
= 10.
(b) Give values different from those in part (a) for which
R
(
m, n
) is false.
(c) Explain why
R
(
m, n
) is true if
m
= 5 and
n
= 10.
(d) Give values different from those in part (c) for which
R
(
m, n
) is true.
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81. Find the truth set of each predicate.
(a) predicate:
6
d
is an integer, domain:
Z
(b) predicate:
6
d
is an integer, domain:
Z
+
(c) predicate: 1
≤
x
2
≤
4, domain:
R
(d) predicate: 1
≤
x
2
≤
4, domain:
Z
82. Let
B
(
x
) be “
−
10
< x <
10.” Find the truth set of
B
(
x
) for each of the
following domains.
(a)
Z
(b)
Z
+
(c) The set of all even integers
Find counterexamples to show that the statements are false.
83.
∀
x
∈
R
, x >
1
x
.
84.
∀
a
∈
Z
,
a
−
1
a
is not an integer.
85.
∀
positive integers
m
and
n
,
m
·
n
≥
m
+
n
.
86.
∀
real numbers
x
and
y
,
√
x
+
y
=
√
x
+
√
y
.
87. Consider the following statement:
∀
basketball players
x
,
x
is tall. Which
of the following are equivalent ways of expressing this statement?
(a) Every basketball player is tall.
(b) Among all the basketball players, some are tall.
(c) Some of all the tall people are basketball players.
(d) Anyone who is tall is a basketball player.
(e) All people who are basketball players are tall.
(f) Anyone who is a basketball player is a tall person.
88. Consider the following statement:
∃
x
∈
R
such that
x
2
= 2. Which of the
following are equivalent ways of expressing this statement?
(a) The square of each real number is 2.
(b) Some real numbers have square 2.
(c) The number
x
has square 2, for some real number
x
.
(d) If
x
is a real number, then
x
2
= 2.
(e) Some real number has square 2.
(f) There is at least one real number whose square is 2.
89. Rewrite the following statements informally in at least two different ways
without using variables or quantifiers.
16
(a)
∀
rectangles
x
,
x
is a quadrilateral.
(b)
∃
a set
A
such that
A
has 16 subsets.
90. Rewrite each of the following statements in the form “
∀
........
x,
........
.
”
(a) All dinosaurs are extinct.
(b) Every real number is positive, negative, or zero.
(c) No irrational numbers are integers.
(d) No logicians are lazy.
(e) The number 2,147,581,953 is not equal to the square of any integer.
(f) The number
−
1 is not equal to the square of any real number.
91. Rewrite each of the following in the form “
∃
........
x
such that
.........
”
(a) Some exercises have answers.
(b) Some real numbers are rational.
92. Let
D
be the set of all students at your school, and let
M
(
s
) be “
s
is a
math major,” let
C
(
s
) be “
s
is a computer science student,” and let
E
(
s
)
be “
s
is an engineering student.” Express each of the following statements
using quantifiers, variables, and the predicates
M
(
s
),
C
(
s
), and
E
(
s
).
(a) There is an engineering student who is a math major.
(b) Every computer science student is an engineering student.
(c) No computer science students are engineering students.
(d) Some computer science students are also math majors.
(e) Some computer science students are engineering students and some
are not.
93. Consider the following statement:
∀
integers
n
, if
n
2
is even then
n
is even.
Which of the following are equivalent ways of expressing this statement?
(a) All integers have even squares and are even.
(b) Given any integer whose square is even, that integer is itself even.
(c) For all integers, there are some whose square is even.
(d) Any integer with an even square is even.
(e) If the square of an integer is even, then that integer is even.
(f) All even integers have even squares.
94. Rewrite the following statement informally in at least two different ways
without using variables or the symbol
∀
or the words “for all.”
∀
real
numbers
x
, if
x
is positive, then the square root of
x
is positive.
95. Rewrite the following statements so that the quantifier trails the rest of
the sentence.
17
(a) For any graph
G
, the total degree of
G
is even.
(b) For any isosceles triangle
T
, the base angles of
T
are equal.
(c) There exists a prime number
p
such that
p
is even.
(d) There exists a continuous function
f
such that
f
is not differentiable.
96. Rewrite each of the following statements in the form “
∀
.........
x,
if
.........
then
......... .
”
(a) All Java programs have at least 5 lines.
(b) Any valid argument with true premises has a true conclusion.
97. Rewrite each of the following statements in the two forms “
∀
.........
x,
if
.........
then
.........
” and “
∀
.........
x,
.........
” (without an if-then).
(a) All equilateral triangles are isosceles.
(b) Every computer science student needs to take data structures.
98. Rewrite the following statements in the two forms “
∃
.........
x
such that
.........
” and “
∃
.........
x
such that
.........
and
......... .
”
(a) Some hatters are mad.
(b) Some questions are easy.
99. The statement “The square of any rational number is rational” can be
rewritten formally as “For all rational numbers
x, x
2
is rational” or as “For
all
x
, if
x
is rational then
x
2
is rational.” Rewrite each of the following
statements in the two forms “
∀
.........
x,
.........
” and “
∀
x,
if
.........
, then
.........
” or in the two forms “
∀
.........
x
and
y,
”
.........
and “
∀
x
and
y,
if
.........
, then
..........
”
(a) The reciprocal of any nonzero fraction is a fraction.
(b) The derivative of any polynomial function is a polynomial function.
(c) The sum of the angles of any triangle is 180
◦
.
(d) The negative of any irrational number is irrational.
(e) The sum of any two even integers is even.
(f) The product of any two fractions is a fraction.
100. Consider the statement “All integers are rational numbers but some ra-
tional numbers are not integers.”
(a) Write this statement in the form “
∀
.........
x,
.........
if
.........
then
.........
,
but
∃
.........
x
such that
......... .
”
(b) Let Ratl(
x
) be “
x
is a rational number” and Int(
x
) be “
x
is an in-
teger.” Write the given statement formally using only the symbols
Ratl(
x
), Int(
x
),
∀
,
∃
,
∧
,
∨
,
∼
, and
→
.
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101. Let the domain of
x
be the set
D
of objects discussed in mathematics
courses, and let
Real
(x) be “
x
is a real number,”
Pos
(x) be “
x
is a positive
real number,”
Neg
(x) be “
x
is a negative real number,” and
Int
(x) be “
x
is an integer.”
(a)
Pos
(0)
(b)
∀
x,
Real
(
x
)
∧
Neg
(
x
)
→
Pos
(
−
x
).
(c)
∀
x,
Int
(
x
)
→
Real
(
x
).
(d)
∃
x
such that
Real
(
x
)
∧ ∼
Int
(
x
).
102. Let the domain of
x
be the set of geometric figures in the plane, and let
Square
(x) be “
x
is a square” and
Rect
(x) be “
x
is a rectangle.”
(a)
∃
x
such that
Rect
(
x
)
∧
Square
(
x
).
(b)
∃
x
such that
Rect
(
x
)
∧ ∼
Square
(
x
).
(c)
∀
x,
Square
(
x
)
→
Rect
(
x
).
103. Let the domain of
x
be the set
Z
of integers, and let
Odd
(x) be “
x
is odd,”
Prime
(x) be “
x
is prime,” and
Square
(x) be “
x
is a perfect square.” (An
integer
n
is said to be a perfect square if, and only if, it equals the square
of some integer. For example, 25 is a perfect square because 25 = 5
2
.)
(a)
∃
x
such that
Prime
(
x
)
∧ ∼
Odd
(
x
).
(b)
∀
x,
Prime
(
x
)
→∼
Square
(
x
).
(c)
∃
x
such that
Odd
(
x
)
∧
Square
(
x
).
104. In any mathematics or computer science text other than this book, find
an example of a statement that is universal but is implicitly quantified.
Copy the statement as it appears and rewrite it making the quantification
explicit. Give a complete citation for your example, including title, author,
publisher, year, and page number.
105. Let
R
be the domain of the predicate variable
x
. Which of the following
are true and which are false? Give counter examples for the statements
that are false.
(a)
x >
2
⇒
x >
1
(b)
x >
2
⇒
x
2
>
4
(c)
x
2
>
4
⇒
x >
2
(d)
x
2
>
4
⇔ |
x
|
>
2
106. Let
R
be the domain of the predicate variables
a, b, c,
and
d
.
Which of
the following are true and which are false? Give counterexamples for the
statements that are false.
(a)
a >
0 and
b >
0
⇒
ab >
0
19
(b)
a <
0 and
b <
0
⇒
ab <
0
(c)
ab
= 0
⇒
a
= 0 or
b
= 0
(d)
a < b
and
c < d
⇒
ac < bd
107. Write a formal negation for each of the following statements:
(a)
∀
fish
x, x
has gills.
(b)
∀
computers
c, c
has a CPU.
(c)
∃
a movie
m
such that
m
is over 6 hours long.
(d)
∃
a band
b
such that
b
has won at least 10 Grammy awards.
108. 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 people are happy.
(c) Some suspicions were substantiated.
(d) Some estimates are accurate.
109. Write a negation for each of the following statements.
(a) Any valid argument has a true conclusion.
(b) Every real number is positive, negative, or zero.
110. Write a negation for each of the following statements.
(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.
111. Informal language is actually more complex than formal language.
For
instance, the sentence “There are no orders from store
A
for item
B
”
contains the words there are.
Is the statement existential?
Write an
informal negation for the statement, and then write the statement formally
using quantifiers and variables.
112. 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.
113.
∀
real numbers
x,
if
x >
3 then
x
2
>
9
.
114.
∀
computer programs
P,
if
P
compiles without error messages, then
P
is correct.
Determine whether the proposed negation is correct. If it is not,
write a correct negation.
20
115. Statement: The sum of any two irrational numbers is irrational.
Proposed negation: The sum of any two irrational numbers is rational.
116. 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.
117. Statement: For all integers
n
, if
n
2
is even then
n
is even.
Proposed negation: For all integers
n
, if
n
2
is even then
n
is not even.
118. Statement: For all real numbers
x
1
and
x
2
, if
x
2
1
=
x
2
2
then
x
1
=
x
2
.
Proposed negation:
For all real numbers
x
1
and
x
2
, if
x
2
1
=
x
2
2
then
x
1
̸
=
x
2
.
119. 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 counterexam-
ples for those statements that are false.
(a)
∀
x
∈
D
, if
x
is odd then
x >
0.
(b)
∀
x
∈
D
, if
x
is less than 0 then
x
is even.
(c)
∀
x
∈
D
, if
x
is even then
x
≤
0.
(d)
∀
x
∈
D
, if the ones digit of
x
is 2, then the tens digit is 3 or 4.
(e)
∀
x
∈
D
, if the ones digit of
x
is 6, then the tens digit is 1 or 2.
Write a negation for each statement.
120.
∀
real numbers
x
, if
x
2
≥
1 then
x >
0.
121.
∀
integers
d
, if
6
d
is an integer then
d
= 3.
122.
∀
x
∈
R
, if
x
(
x
+ 1)
>
0 then
x >
0 or
x <
−
1.
123.
∀
n
∈
Z
, if
n
is prime then
n
is odd or
n
= 2.
124.
∀
integers
a, b
and
c
, if
a
−
b
is even and
b
−
c
is even, then
a
−
c
is even.
125.
∀
integers
n
, if
n
is divisible by 6, then
n
is divisible by 2 and
n
is divisible
by 3.
126. If the square of an integer is odd, then the integer is odd.
127. If a function is differentiable then it is continuous.
128. 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.
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(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.
129. Each of the following statements is true. In each case write the converse
of the 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 any odd integer, then 2
m
is even.
(c) If two circles intersect in exactly two points, then they do not have
a common center.
130. Write the contrapositive for each of the following statements.
(a) If
n
is prime, then
n
is not divisible by any prime number between
1 and
√
n
strictly. (Assume that
n
is a fixed integer that is greater
than 1.)
(b) If
A
and
B
do not have any elements in common, then they are
disjoint. (Assume that
A
and
B
are fixed sets.)
131. Give an example to show that a universal conditional statement is not
logically equivalent to its inverse.
Some of the arguments are valid by universal modus ponens
or universal modus tollens; others are invalid and exhibit the
converse or the inverse error.
State which are valid and which
are invalid. Justify your answers.
132. All healthy people eat an apple a day.
Keisha eats an apple a day.
∴
Keisha is a healthy person.
133. All freshmen must take writing.
Caroline is a freshman.
∴
Caroline must take writing.
134. All healthy people eat an apple a day.
Herbert is not a healthy person.
∴
Herbert does not eat an apple a day.
135. If a product of two numbers is 0, then at least one of the numbers is 0.
For a particular number
x
, neither (2
x
+ 1) nor (
x
−
7) equals 0.
∴
The product (2
x
+ 1)(
x
−
7) is not 0.
136. All cheaters sit in the back row.
Monty sits in the back row.
∴
Monty is a cheater.
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137. All honest people pay their taxes.
Darth is not honest.
∴
Darth does not pay his taxes.
138. For all students
x
, if
x
studies discrete mathematics, then
x
is good at
logic.
Tarik studies discrete mathematics.
∴
Tarik is good at logic.
139. If compilation of a computer program produces error messages, then the
program is not correct.
Compilation of this program does not produce error messages.
∴
This program is correct.
140. Any sum of two rational numbers is rational.
The sum
r
+
s
is rational.
∴
The numbers
r
and
s
are both rational.
141. If a number is even, then twice that number is even.
The number 2
n
is even, for a particular number
n
.
∴
The particular number
n
is even.
142. If an infinite series converges, then the terms go to 0.
The terms of the infinite series
∑
∞
n
=1
1
n
go to 0.
∴
The infinite series
∑
∞
n
=1
1
n
converges.
143. If an infinite series converges, then its terms go to 0.
The terms of the infinite series
∑
∞
n
=1
n
n
+1
do not go to 0.
∴
The infinite series
∑
∞
n
=1
n
n
+1
does not converge.
144. Rewrite the statement “No good cars are cheap” in the form “
∀
x,
if
P
(
x
) then
∼
Q
(
x
)
.
” Indicate whether each of the following arguments is valid or invalid,
and justify your answers.
(a) No good car is cheap.
A Rimbaud is a good car.
∴
A Rimbaud is not cheap.
(b) No good car is cheap.
A Simbaru is not cheap.
∴
A Simbaru is a good car.
(c) No good car is cheap.
A VX Roadster is cheap.
∴
A VX Roadster is not good.
(d) No good car is cheap.
An Omnex is not a good car.
∴
An Omnex is cheap.
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