3.1
Statements and Quantifiers
99
The many relationships among special sets of numbers can be expressed using
universal and existential quantifiers. Earlier we introduced sets of numbers that are
studied in algebra, and we repeat these in the box that follows.
Sets of Numbers
Natural or Counting numbers 1, 2, 3, 4,…
Whole numbers 0, 1, 2, 3, 4,…
Integers …, 3, 2, 1, 0, 1, 2, 3,…
Rational numbers p
q p and q are integers, and q 0
(Some examples of rational numbers are 3
5, 7
9, 5, and 0. Any
rational number may be written as a terminating decimal number, like
.25 or a repeating decimal number, like .666….)
Real numbers x x is a number that can be written as a decimal
Irrational numbers x x is a real number and x cannot be written as
a quotient of integers
3
4, and . Decimal
(Some examples of irrational numbers are 2, representations of irrational numbers never terminate and never repeat.
EXAMPLE 6
Decide whether each of the following statements about sets of
numbers involving a quantifier is true or false.
(a) There exists a whole number that is not a natural number.
Because there is such a whole number (it is 0), this statement is true.
(b) Every integer is a natural number.
This statement is false, because we can find at least one integer that is not a
natural number. For example, 1 is an integer but is not a natural number.
(There are infinitely many other choices we could have made.)
(c) Every natural number is a rational number.
Since every natural number can be written as a fraction with denominator 1, this
statement is true.
(d) There exists an irrational number that is not real.
In order to be an irrational number, a number must first be real (see the box).
Therefore, since we cannot give an irrational number that is not real, this statement is false. (Had we been able to find at least one, the statement would have
then been true.)
3.1
EXERCISES
Decide whether each of the following is a statement or
is not a statement.
3. Listen, my children, and you shall hear of the midnight ride of Paul Revere.
1. December 7, 1941, was a Sunday.
4. Yield to oncoming traffic.
2. The ZIP code for Manistee, MI, is 49660.
5. 5 8 13
and
431
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100
CHAPTER 3
6. 5 8 12
Introduction to Logic
or 4 3 2
7. Some numbers are negative.
8. Andrew Johnson was president of the United States
in 1867.
9. Accidents are the main cause of deaths of children
under the age of 8.
10. Star Wars: Episode I—The Phantom Menace was the
top-grossing movie of 1999.
Give a negation of each inequality. Do not use a slash
symbol.
33. y 12
34. x 6
35. q 5
36. r 19
37. Try to negate the sentence “The exact number of
words in this sentence is ten” and see what happens.
Explain the problem that arises.
38. Explain why the negation of “r 4” is not “r 4.”
11. Where are you going today?
12. Behave yourself and sit down.
13. Kevin “Catfish” McCarthy once took a prolonged
continuous shower for 340 hours, 40 minutes.
14. One gallon of milk weighs more than 4 pounds.
Let p represent the statement “She has green eyes” and
let q represent the statement “He is 48 years old.”
Translate each symbolic compound statement into
words.
39. p
40. q
Decide whether each of the following statements is
compound.
41. p q
42. p q
43. p q
44. p q
15. I read the Chicago Tribune and I read the New York
Times.
45. p q
46. p q
47. p q
48. p q
16. My brother got married in London.
17. Tomorrow is Sunday.
18. Dara Lanier is younger than 29 years of age, and so
is Teri Orr.
19. Jay Beckenstein’s wife loves Ben and Jerry’s ice
cream.
20. The sign on the back of the car read “California or
bust!”
21. If Julie Ward sells her quota, then Bill Leonard will
be happy.
Let p represent the statement “Chris collects videotapes” and let q represent the statement “Jack plays the
tuba.” Convert each of the following compound statements into symbols.
49. Chris collects videotapes and Jack does not play the
tuba.
50. Chris does not collect videotapes or Jack does not
play the tuba.
22. If Mike is a politician, then Jerry is a crook.
51. Chris does not collect videotapes or Jack plays the
tuba.
Write a negation for each of the following statements.
52. Jack plays the tuba and Chris does not collect videotapes.
23. Her aunt’s name is Lucia.
24. The flowers are to be watered.
25. Every dog has its day.
26. No rain fell in southern California today.
27. Some books are longer than this book.
28. All students present will get another chance.
29. No computer repairman can play blackjack.
30. Some people have all the luck.
31. Everybody loves somebody sometime.
32. Everyone loves a winner.
53. Neither Chris collects videotapes nor Jack plays the
tuba.
54. Either Jack plays the tuba or Chris collects videotapes, and it is not the case that both Jack plays the
tuba and Chris collects videotapes.
55. Incorrect use of quantifiers often is heard in everyday language. Suppose you hear that a local electronics chain is having a 30% off sale, and the radio
advertisement states “All items are not available in
all stores.” Do you think that, literally translated, the
ad really means what it says? What do you think really is meant? Explain your answer.
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3.1
Statements and Quantifiers
56. Repeat Exercise 55 for the following: “All people
don’t have the time to devote to maintaining their
cars properly.”
Decide whether each statement in Exercises 65–74
involving a quantifier is true or false.
Refer to the groups of art labeled A, B, and C, and
identify by letter the group or groups that are satisfied
by the given statements involving quantifiers.
66. Every natural number is an integer.
101
65. Every whole number is an integer.
67. There exists a rational number that is not an integer.
68. There exists an integer that is not a natural number.
69. All rational numbers are real numbers.
70. All irrational numbers are real numbers.
71. Some rational numbers are not integers.
A
72. Some whole numbers are not rational numbers.
73. Each whole number is a positive number.
74. Each rational number is a positive number.
75. Explain the difference between the following
statements:
B
All students did not pass the test.
Not all students passed the test.
C
57. All pictures have frames.
58. No picture has a frame.
76. The statement “For some real number x, x 2 0” is
true. However, your friend does not understand why,
since he claims that x 2 0 for all real numbers x
(and not some). How would you explain his misconception to him?
77. Write the following statement using “every”: There
is no one here who has not done that at one time or
another.
59. At least one picture does not have a frame.
60. Not every picture has a frame.
61. At least one picture has a frame.
62. No picture does not have a frame.
63. All pictures do not have frames.
64. Not every picture does not have a frame.
78. Only one of the following statements is true. Which
one is it?
A. For some real number x, x 0.
B. For all real numbers x, x 3 0.
C. For all real numbers x less than 0, x 2 is also less
than 0.
D. For some real number x, x 2 0.
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