Download 1 false true true true false false 2. If x2 = 9, then x could equal +3 or

Chapter 1, Part I (pages 2-5)
2.
inductive reasoning
4.
inductive reasoning
6.
inductive reasoning
8.
deductive reasoning
14. This statement could be rewritten as,
“If a person lives in Illinois, then he or
she lives in Chicago,” and it is false
because there are some people that
live in Illinois that do not live in
Chicago.
10. deductive reasoning
12. 1 and –1
14. When you add zero to a number, the
answer is equal to the original number.
16. It will rain today in the town Vanessa
has just moved to.
18. Martha will hear a dog barking as she
walks by the house at 123 Main Street.
20. Carrie will receive a paycheck
tomorrow.
22. (a), (b), (d)
Chapter 1, Part II (pages 9-13)
2
2.
If x = 9, then x could equal +3 or –3.
Therefore, this statement is false .
4.
This statement is true .
6.
Since the part of the statement that
follows the “if” is false, the statement
is true .
8.
Chapter 1, Part II (pages 9-13) continued
16. Since the part of this statement that
follows the “if” is true and the part of
the statement that follows the “then”
is false, this statement is false .
18. Since the part of this statement that
follows the “if” is false, it does not
matter what the part of the statement
that follows the “then” says; the
statement is true .
20. This statement can be rewritten as, “If
a figure has four sides, then it is a
rectangle.” The figure below shows a
counterexample to this statement,
and so the statement must be . false .
6 ft
This statement could be rewritten as,
“If there is a unicorn on Saturn, then it
loves pepperoni pizza.” Since the
part of this statement that follows the
“if” is false, the statement is true .
10. This statement could be rewritten as,
“If a number is greater than 7, then it
is greater than 10.” The number 8 is
a counterexample that proves this
statement wrong, and so this
statement must be false .
12. This statement could be rewritten as,
“If a person lives in Chicago, then he
or she may live in Illinois,” and so it is
true .
3 ft
2 ft
5 ft
22. Example: 9 is an odd number, but it
is not prime
24. Example: 0.1 is a real number, but
0.12 is not greater than 0.1
26. Example: The figure shown below
has four sides, but it is not a square.
15 cm
10 cm
15 cm
10 cm
28. Example: 3.5 > 3, but 3.52 is not
greater than 15
30. Example: 24 is divisible by both 6
and 12
© 2011 A+ Education Services
1
Chapter 1, Part II (pages 9-13) continued
32. Let p represent the statement, “You do
your homework,” let q represent the
statement, “You get better grades,”
and let r represent the statement, “You
will get in into a better college.” Then
you can use the Law of Syllogism to
say that the following statement is
true.
If you do your homework, then you
will get into a better college.
Chapter 1, Part II (pages 9-13) continued
40. No conclusion is possible.
(Statement (a) is a statement in the
form If p, then q, and statement (b) tells
us that q is true, but neither the Law of
Detachment nor the Law of Syllogism
allow us to make a conclusion based
on these statements.)
42. No conclusion is possible.
(Statement (a) is a statement in the
form If p, then q, and statement (b) is a
statement in the form If p, then r, but
neither the Law of Detachment nor the
Law of Syllogism allow us to make a
conclusion based on these
statements.)
34. Let p represent the statement, “An
animal is a bird,” and let q represent
the statement, “It has feathers.” Then
(b) tells us that the statement p is true
for a cardinal, and so you can use the
Law of Detachment to say that the
following statement is true.
A cardinal has feathers.
Chapter 1, Part III (pages 17-23)
2.
36. No conclusion is possible.
(Statement (a) is a statement in the
form If p, then q, and statement (b) tells
us that q is true, but neither the Law of
Detachment nor the Law of Syllogism
allow us to make a conclusion based
on these statements.)
38. Statement (b) can be rewritten as, “If a
number is a whole number, then it is
an integer.”
Let p represent the statement, “A
number is a natural number,” let q
represent the statement, “A number is
a whole number,” and let r represent
the statement, “A number is an
integer.” Then you can use the Law of
Syllogism to say that the following
statement is true.
(a) and (b) are both correct
negations of this statement
To see why (c) and (e) are not correct
negations of this statement:
Suppose that m = 3 and n = 22.
Then the original statement, (c), and
(e) would all be true. Since the
original statement and its negation
cannot both be true at the same time,
(c) and (e) cannot be correct
negations of the original statement.
To see why (d) and (f) are not correct
negations of the statement:
Suppose that m and n both equal 12.
Then the original statement, (d), and
(f) would all be true. Since the
original statement and its negation
cannot both be true at the same time,
(d) and (f) cannot be correct
negations of the original statement.
If a number is a natural number, then
it is an integer.
2
© 2011 A+ Education Services
Chapter 1, Part III (pages 17-23) continued
Chapter 1, Part III (pages 17-23) continued
4.
(a) and (d) are both correct
negations of this statement
(b) is not a correct negation because,
if the light is off, then both the original
statement and (b) would be false
(c) is not a correct negation because,
if the light is dim, then both the
original statement and (c) would be
false
(e) is not a correct negation because
it is possible for both the original
statement and statement (e) to be
true (or false) at the same time
14. The original statement is true. (Note
that this statement makes no claim as
to what happens if 3 • 4 does not
equal 25.)
The converse is, “If 5 + 1 = 2, then
3 • 4 = 25,” and it is true. (Note
that this statement makes no claim
as to what happens if 5 + 1 does
not equal 2.)
The inverse is, “If 3 • 4 ≠ 25, then
5 + 1 ≠ 2,” and it is true.
The contrapositive is, “If 5 + 1 ≠ 2,
then 3 • 4 ≠ 25,” and it is true.
6.
(c) and (d) are correct negations of
this statement
To see why (a) and (b) are not
correct negations of the statement:
Suppose that the bell pepper is
yellow. Then the original statement,
as well as statements (a), and (b)
would all be false, and a statement
and its negation cannot both be false
at the same time.
(e) is not a correct negation because
it is possible for both the original
statement and statement (e) to be
true (or false) at the same time
16. Note that the original statement can
be rewritten as, “If a figure is a
square, then it has four sides,” and it
is true.
The converse is, “If a figure has four
sides, then it is a square,” and it is
false.
The inverse is, “If a figure is not a
square, then it does not have four
sides,” and it is false.
The contrapositive is, “If a figure does
not have four sides, then it is not a
square,” and it is true.
8.
(d) is a correct negation of the
statement
10. (d) and (f) are correct negations of
this statement
12. The original statement is false.
The converse is, “If x > 5, then x = 3,”
and it is false.
The inverse is, “If x ≠ 3, then x ≤ 5,”
and it is false.
The contrapositive is, “If x ≤ 5, then
x ≠ 3,” and it is false.
18. The original statement can be
rewritten as, “If a number is divisible
by 2, then it is divisible by 4,” and it is
false.
The converse is, “If a number is
divisible by 4, then it is divisible by 2,”
and it is true.
The inverse is, “If a number is not
divisible by 2, then it is not divisible by
4,” and it is true.
The contrapositive is, “If a number is
not divisible by 4, then it is not
divisible by 2,” and it is false.
© 2011 A+ Education Services
3
Chapter 1, Part III (pages 17-23) continued
20. The converse is, “If you don’t have to
remember anything, then you told the
truth.”
The inverse is, “If you don’t tell the
truth, then you have to remember
something.”
The contrapositive is, “If you have to
remember something, then you didn’t
tell the truth.”
22. The original statement is telling us
that, if we ever find a pacadam, then
it will be brackle. Therefore, only
statements (b), (d), (e), and (g) are
equivalent to the original. (Note that
statement (b) is the contrapositive of
the original statement.)
24. Only statement (c) is equivalent to
the original statement. (Note that (c)
is the contrapositive of the original
statement.)
26. The statement is telling us that, if we
find a molecule of water, then it is
made up of hydrogen and oxygen.
Thus, statements (b) and (c) are
equivalent to the original statement.
(Note that (b) is the contrapositive of
the original statement.)
28. Billy may be in Wyoming.
Chapter 1, Part IV (pages 25-28)
2.
4
This statement is saying (1) If x = y,
then x + 3 = y + 3, and (2) If x + 3 = y + 3,
then x = y. Since both of these
statements are true, the original
statement must also be true .
Chapter 1, Part IV (pages 25-28) continued
4.
This statement is saying (1) If an
insect is a mosquito, then it has
wings, and (2) If an insect has wings,
then it is a mosquito. Since
statement (2) is false, the original
statement is also false .
6.
This statement is saying (1) If | –5| = 2,
then 7 > 2, and (2) If 7 > 2, then
| –5| = 2. Both of these statements are
true (note that statement (1) makes no
claim about what happens if | –5| ≠ 2,
and statement (2) makes no claim
about what happens if 7 is not greater
than 2), and so the original statement
is true .
8.
This statement is saying (1) If x = 3,
then x < 4, and (2) If x < 4, then x = 3.
Since statement (2) is false, the
original statement must also be false .
10. This statement is saying (1) If today is
Thursday, then tomorrow is Friday,
and (2) If tomorrow is Friday, then
today is Thursday. Both of these
statements are true, and so the
original statement must be true .
12. The original statement says (1) If
Marlene eats a piece of the pizza,
then it has Italian sausage on it, and
(2) If the pizza has Italian sausage on
it, then Marlene will eat a piece of the
pizza. Therefore, (a), (b), (c), and (d)
are all true. (Note that (c) is the
contrapositive of statement (2), and (d)
is the contrapositive of statement (1).)
© 2011 A+ Education Services
Chapter 1, Part IV (pages 25-28) continued
Chapter 1, Part V (pages 32-40)
14. The original statement is saying (1) If
Jake does the dishes, then his sister
will help him, and (2) If his sister
helps him, then Jake will do the
dishes. Therefore, (c), (d), (e), and (f)
are all true. (Note that (d) is the
contrapositive of statement (2), and (f)
is the contrapositive of statement (1).)
Note: For each of the problems in this
section, there are many other proofs
that are equally correct.
2.
Statements
Reasons
16. The original statement is saying (1) If
a quadrilateral is a parallelogram,
then its diagonals bisect each other,
and (2) If the diagonals of a
quadrilateral bisect each other, then
the quadrilateral is a parallelogram.
Thus, statements (a), (c), (d), and (e)
are all true. (Note that (c) is the
contrapositive of statement (1), and (e)
is the contrapositive of statement (2).)
18. The original statement is saying (1) In
Italian, if an object it called a matita,
then it is a pencil, and (2) If an object
is a pencil, then it is called a matita in
Italian. Therefore, (a), (b), (d) and (e)
are all true. (Note that (b) is the
contrapositive of statement (1), and (e)
is the contrapositive of statement (2).)
1. 7a – 2 = 3(2a – 4)
1. Given
2. 7a – 2 = 6a – 12
2. Distributive
Property
3. a – 2 = –12
3. Subtraction
Property of Equality
4. a = –10
4. Addition
Property of Equality
4.
Statements
Reasons
1.
p
+3=7
5
1. Given
2.
p
=4
5
2. Subtraction
Property of Equality
3. p = 20
3. Multiplication
Property of Equality
6.
Statements
Reasons
1. 2k + 3 = 5 – 2(k + 4) 1. Given
2. 2k + 3 = 5 – 2k – 8
2. Distributive
Property
3. 2k + 3 = –3 – 2k
3. Substitution
Property of Equality
4. 4k + 3 = –3
4. Addition
Property of Equality
5. 4k = – 6
5. Subtraction
Property of Equality
3
2
6. Division
Property of Equality
6. k = −
© 2011 A+ Education Services
5
Chapter 1, Part V (pages 32-40) continued
8.
Statements
14.
Statements
Reasons
1. 2(d – 3) + 1 = d – 5 1. Given
2. 2d – 6 + 1 = d – 5
Chapter 1, Part V (pages 32-40) continued
2. Distributive
Property
3. 2d – 5 = d – 5
3. Substitution
Property of Equality
4. 2d = d
4. Addition Property
of Equality
5. d = 0
5. Subtraction
Property of Equality
1.
5 2u
1
+
>
6 3
2
2.
2u
1
>–
3
3
3. 2u > –1
4. u > –
1
2
Reasons
1. Given
2. Subtraction
Property of
Inequality
3. Multiplication
Property of
Inequality
4. Division
Property of
Inequality
10.
Statements
Reasons
1. 6 – 3m > 4
1. Given
16.
Statements
2. –3m > –2
2. Subtraction
Property of
Inequality
1. 2x + 3y = –17 and
x – 4y = 8
1. Given
3. m < 2
3
3. Division Property
of Inequality
2. x = 8 + 4y
2. Addition
Property of Equality
3. 2(8 + 4y) + 3y = –17
3. Substitution
Property of Equality
Reasons
12.
Statements
Reasons
1. 5 + 2(3 – c) ≤ 4c
1. Given
4. 16 + 8y + 3y = –17
4. Distributive
Property
2. 5 + 6 – 2c ≤ 4c
2. Distributive
Property
5. 16 + 11y = –17
5. Distributive
Property
3. Substitution
Property of
Inequality
6. 11y = –33
6. Subtraction
Property of Equality
3. 11 – 2c ≤ 4c
4. 11 ≤ 6c
4. Addition Property
of Inequality
7. y = –3
7. Division
Property of Equality
11
≤c
6
5. Division Property
of Inequality
8. x – 4(–3) = 8
8. Substitution
Property of Equality
9. x + 12 = 8
9. Substitution
Property of Equality
10. x = – 4
10. Subtraction
Property of Equality
5.
6. c ≥
6
11
6
6. Symmetric
Property of
Inequality
© 2011 A+ Education Services
Chapter 1, Part V (pages 32-40) continued
Chapter 1, Part V (pages 32-40) continued
Reasons
22.
Statements
1. 8x – y = 4 and
5x + 2y = – 8
1. Given
1.
2x
–4=1
3
1. Given
2. 8x = 4 + y
2. Addition Property
of Equality
2. Assume that x ≠
15
. 2. Assumption
2
3. 8x – 4 = y
3. Subtraction
Property of Equality
3. 2x ≠ 15
4. 5x + 2(8x – 4) = – 8
4. Substitution
Property of Equality
18.
Statements
5. 5x + 16x – 8 = – 8
5. Distributive
Property
6. 21x – 8 = – 8
6. Distributive
Property
7. 21x = 0
7. Addition Property
of Equality
8. x = 0
8. Division Property
of Equality
9. 8(0) – y = 4
9. Substitution
Property of Equality
10. –y = 4
10. Substitution
Property of Equality
11. y = – 4
4.
2x
≠5
3
5.
2x
–4≠1
3
6. x =
Reasons
1. 2(b + 5) = 12
1. Given
2. Assume that b ≠ 1. 2. Assumption
3. b + 5 ≠ 6
3. Addition Property
of Inequality
4. 2(b + 5) ≠ 12
4. Multiplication
Property of Inequality
5. b = 1
5. Contradiction
3. Multiplication
Property of
Inequality
4. Division
Property of
Inequality
5. Subtraction
Property of
Inequality
15
2
6. Contradiction
24.
Statements
Reasons
1. 3z + 1 < 5
1. Given
2. Assume that z </
4
.
3
2. Assumption
3. 3z </ 4
3. Multiplication
Property of
Inequality
4. 3z + 1 </ 5
4. Addition
Property of
Inequality
11. Division
Property of Equality
20.
Statements
Reasons
5. z <
4
3
© 2011 A+ Education Services
5. Contradiction
7