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Finite Mathematics
Chapter 7
Name ________________________________ Date ______________ Class ____________
Section 7-1 Logic
Goal: To convert statements in symbolic logic form and create truth tables
p
T
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q
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Øp
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STATEMENT:
CONVERSE:
INVERSE:
CONTRAPOSITIVE:
p Úq
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F
p Ùq
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F
p® q
q® p
Øp ® Øq
Øq ® Øp
p® q
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T
If p then q
If q then p
If not p then not q
If not q then not p
In Problems 1–10, write each of the following statements in symbolic form.
1. If I go to bed early, then I will wake up early.
Let p: I go to bed early and q: I will wake up early, then symbolically p ® q.
2. George will play cards or he will play pool.
Let p: George will play cards and q: he will play pool, then symbolically p Ú q.
3. The cake was not burned and very tasteful.
Let p: The cake was burned and q: The cake was very tasteful, then symbolically
Øp Ù q.
4. Margaret did not pass physics.
Let p: Margaret passed physics, then symbolically Øp.
5. Hank did not come home for spring break and he had a wonderful time.
Let p: Hank came home for spring break and q: he had a wonderful time, then
symbolically Øp Ù q
7-1
Copyright © 2015 Pearson Education, Inc.
Finite Mathematics
Chapter 7
6. The committee decided to table the motion and bring it up at the next meeting.
Let p: The committee decided to table the motion and q: The committee will bring it
up at the next meeting, then symbolically p Ù q
7. Carol decided to take a math class or a psychology class.
Let p: Carol decided to take a math class and q: Carol decided to take a psychology
class, then symbolically p Ú q
8. Nathan did not agree to go to the movies.
Let p: Nathan agreed to go to the movies, then symbolically Øp
9. If Mary and John decide on a restaurant at which to eat dinner, then they can make
reservations.
Let p: Mary and John decide on a restaurant at which to eat and q: they can make
reservations, then symbolically p ® q
10. If Peggy does not go to work today, then she will have to take a personal day from
work.
Let p: Peggy goes to work today and q: she will have to take a personal day from
work, then symbolically Øp ® q
In Problems 11–14, state the converse, inverse, and contrapositive of each of the following
statements.
11. If today is Sunday, then yesterday was Saturday.
Converse: If yesterday was Saturday, then today is Sunday.
Inverse: If today is not Sunday, then yesterday was not Saturday.
Contrapositive: If yesterday was not Saturday, then today is not Sunday.
12. If 2 + 3 = 4 , then 8 + 9 = 10 .
Converse: If 8 + 9 = 10 , then 2 + 3 = 4 .
Inverse: If 2 + 3 ¹ 4 , then 8 + 9 ¹ 10 .
Contrapositive: If 8 + 9 ¹ 10 , then 2 + 3 ¹ 4 .
7-2
Copyright © 2015 Pearson Education, Inc.
Finite Mathematics
Chapter 7
13. If two lines intersect, then the lines are not parallel.
Converse: If two lines are not parallel, then the lines intersect.
Inverse: If two lines do not intersect, then the lines are parallel.
Contrapositive: If two lines are parallel, then the lines do not intersect.
14. If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.
Converse: If a parallelogram is a rectangle, then one angle is a right angle.
Inverse: If one angle of a parallelogram is not a right angle, then the parallelogram is
not a rectangle.
Contrapositive: If a parallelogram is not a rectangle, then one angle is not a right
angle.
In problems 15–20, construct a truth table for each of the following propositions.
(Note: See below for solutions)
15. Ø (Øp Ù q )
16. q Ú (Øp Ù q )
17. Øp ÚØq
18.
p Ú( p ® q )
20.
( p Ùq) ® ( p Úq)
19.
(Øp ® q ) Ú(Øq ® p )
Solutions:
15.
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Ø ( Øp
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16.
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( Øp
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Ù
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Finite Mathematics
17.
18.
p
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Øp
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Chapter 7
Øq
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( p ® q)
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19.
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q ( Øp
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( p Ùq)
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( p Úq)
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p)
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20.
7-4
Copyright © 2015 Pearson Education, Inc.
Finite Mathematics
Chapter 7
Name ________________________________ Date ______________ Class ____________
Section 7-2 Sets
Goal: To find the unions, intersections, and complements of sets using Venn diagrams and
sets in roster form.
A È B = {x | x Î A or x Î B}
Definitions: Union
Intersection: A Ç B = {x | x Î A and x Î B}
Complement: A¢= {x Î U | x Ï A}
In problems 1 – 10 write the resulting set using the listing method.
1.
{3, 6, 7,8,11,12} Ç {3,5, 6,10,11,14}
{a, b, c, d , e} Ç {a, e, i, o, u}
{a, e}
5.
{3,5, 6,9,12} È {4, 6,8,9,14}
{3, 4,5, 6,8,9,12,14}
{3, 6,11}
3.
2.
4.
{a, b, c, d , e, f , g} È {a, e, i, o, u}
{a, b, c, d , e, f , g , i, o, u}
{1, 2,3, 4,5, 6, 7,8,9} Ç {prime numbers < 10}
{2,3,5, 7}
6.
{1, 2,3, 4,5, 6, 7,8,9} È {prime numbers < 10}
{1, 2,3, 4,5, 6, 7,8,9}
7. For U = {a, e, i, o, u} and A = {a, e} , find A¢.
A¢= {i, o, u}
8. For U = {1, 2,3, 4,5, 6, 7,8,9,10} and A = {1,3,5, 7,9}, find A¢.
A¢= {2, 4, 6,8,10}
7-5
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Finite Mathematics
Chapter 7
9. For U = {prime numbers < 20} and A = {3, 7,11,17} , find A¢.
A¢= {2,5,13,19}
10. For U = {positive integers < 15} and A = {positive integers divisible by 3 and < 15},
find A¢.
A¢= {1, 2, 4,5, 7,810,11,13,14}
11. If P = { p} , list all the subsets of P.
{ p}, { }
12. If T = {a, b} , list all the subsets of T.
{a, b}, {a}, {b}, { }
13. If M = {a, b, c} , list all the subsets of M.
{a, b, c}, {a, b}, {a, c}, {b, c}, {a}, {b}, {c}, { }
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Finite Mathematics
Chapter 7
By referring to the Venn Diagram in Problems 14–16 find how many elements are in the
indicated sets.
a) U
e)
b)
( A È B )¢
U¢
c)
f) A Ç B ¢
AÇ B
d)
AÈ B
g) A¢Ç B
h)
A¢Ç B ¢
14.
a.
b.
c.
d.
e.
f.
g.
h.
80 + 20 + 50 +150 = 300
0
20
80 + 20 + 50 = 150
300 - 150 = 150
80
50
150
7-7
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Finite Mathematics
Chapter 7
15.
a.
b.
c.
d.
e.
f.
g.
h.
25 +15 + 30 + 30 = 100
0
15
25 +15 + 30 = 70
100 - 70 = 30
25
30
30
a.
b.
c.
d.
e.
f.
g.
h.
21 + 3 + 46 + 48 = 118
0
3
21 + 3 + 46 = 70
118 - 70 = 48
21
46
48
16.
7-8
Copyright © 2015 Pearson Education, Inc.
Finite Mathematics
Chapter 7
Name ________________________________ Date ______________ Class ____________
Section 7-3 Basic Counting Principles
Goal: To solve problems using the basic counting principles
Theorem: Addition Principle (for counting)
For any two sets:
n( A È B) = n( A) + n( B) - n( A Ç B)
For two disjoint sets: n( A È B) = n( A) + n( B)
1.
a) Determine the number of ways that the elements of the set {1, 2,3} can be
arranged by listing the possible outcomes.
abc, acb, bac, bca, cab, cba 6 outcomes
b) Determine the number of ways that the elements of the set {1, 2,3} can be
arranged by using a tree diagram.
3
2
1
3
2
3
1
2
3
1
2
1
3
2
1
c) Determine the number of ways that the elements of the set {1, 2,3} can be
arranged by using the multiplication principle.
3 ×2 ×1 = 6
7-9
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Finite Mathematics
Chapter 7
2. Carol picks out a birthday gift for her younger brother and decides to have it gift-wrapped
by the store. She has a choice of 3 wrapping papers (plaid, stripes, or checked). She
must then decide on a ribbon. She has 3 choices for ribbon (red, green, or yellow).
a) Determine the number of ways that Carol can have the gift wrapped by using a
tree diagram.
R
P
G
Y
R
S
G
9 ways
Y
R
C
G
Y
b) Determine the number of ways that Carol can have the gift wrapped by using the
multiplication principle.
3 ×3 = 9
3. Joe and Fred go out to dinner at a local restaurant. On the menu are listed four different
salads and seven different entrees. Joe is not feeling very hungry and decides that he
will have either a salad or an entrée. Fred, however, is ravenous and decides he will
have both a salad and an entrée.
a) How many choices does Joe have if he chooses a salad or an entrée?
4 + 7 = 11choices
b) How many choices does Fred have if he chooses both a salad and an entrée?
4 ´ 7 = 28 choices
7-10
Copyright © 2015 Pearson Education, Inc.
Finite Mathematics
Chapter 7
4. Natalie and Maria go shopping for blouses and capri’s. Natalie has just received her
paycheck and plans to buy both a blouse and a pair of capri’s. Maria has to make her
car payment out of her paycheck and can only afford to buy a blouse or capri’s. The
store they like to shop in has six styles of blouses and eight styles of capri’s. How
many choices does Natalie have? How many choices does Maria have?
Natalie:
6 ´ 8 = 48 choices
Maria:
6 + 8 = 14 choices
5. A survey was done by the local humane society at a shopping mall. The survey asked 200
shoppers if they owned a dog or a cat. The survey found that 80 people owned a cat,
55 people owned a dog, and 23 people owned both a cat and a dog. How many
people in the survey owned neither a dog or a cat? How many people owned a dog
but not a cat?
200 - 57 - 23 - 32 = 88 people did not own a cat or dog.
32 people owned a dog but not a cat
6. A survey was done on a college campus to determine how many students owned a cell
phone. Of the 140 students surveyed, 95 students owned a cell phone, 65 students
had a land line, and 25 students had both a cell phone and a land line. How many
students surveyed had neither a cell phone nor a land line? How many students had a
cell phone but not a land line?
140 - 70 - 25 - 40 = 5 people did not own cell phone or a land line.
70 people owned a cell phone but did not have a land line.
7-11
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Finite Mathematics
Chapter 7
7-12
Copyright © 2015 Pearson Education, Inc.
Finite Mathematics
Chapter 7
Name ________________________________ Date ______________ Class ____________
Section 7-4 Permutations and Combinations
Goal: To solve problems using permutations and combinations
Permutation:
Pn,r = n(n - 1)(n - 2)
Combination:
Cn,r =
Pn,r
r!
=
(n - r + 1) or Pn,r =
n!
(n - r )!
n!
r !(n - r )!
In Problems 1–6, evaluate the permutation or combination.
1. P7,3
P7,3 =
7!
= 210
(7 - 3)!
4. C7,3
C7,3 =
7!
= 35
3!(7 - 3)!
2. P8,5
P8,5 =
3. P10,4
8!
= 6720
(8 - 5)!
5. C8,5
C8,5 =
P10,4 =
10!
= 5040
(10 - 4)!
6. C10,4
8!
= 56
5!(8 - 5)!
C10,4 =
10!
= 210
4!(10 - 4)!
In Problems 7–12 would you consider the following to be a permutation, a combination, or
neither?
7. The number of ways eight people can stand in a line for tickets.
Permutation because order matters
8. The number of ways letters can be assigned for a car license plate.
Permutation because order matters
9. The number of ways students can be chosen to serve on a committee.
Combination because order does not matter
7-13
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Finite Mathematics
Chapter 7
10. The number of ways students can be assigned offices on a committee.
Permutation because order matters
11. The number of ways a 5–card hand can be dealt from a standard deck of cards.
Combination because order does not matter
12. The number of ways nine students can register for a class.
Combination because order does not matter
13. In a mathematics club six people are running for the offices of president, vice-president,
and secretary/treasurer. The student with the most votes will be president, the student with
the next highest number of votes will be vice president, and the third highest number of votes
secretary/treasurer. How many ways can the offices be filled?
Since the order matters, the problem is a permutation. Therefore the answer will be:
P6,3 =
6!
= 120
(6 - 3)!
14. How many ways can 10 people line up in a cafeteria line?
Since the order matters, the problem is a permutation. Therefore the answer will be:
P10,10 =
10!
= 3, 628,800
(10 - 10)!
15. How many ways can 6 people line up in a cafeteria line?
Since the order matters, the problem is a permutation. Therefore the answer will be:
P6,6 =
6!
= 720
(6 - 6)!
16. How many ways can six people be chosen from a group of 12 to attend a conference?
Since the order does not matter, the problem is a combination. Therefore the answer
will be:
C12,6 =
12!
= 924
6!(12 - 6)!
7-14
Copyright © 2015 Pearson Education, Inc.
Finite Mathematics
Chapter 7
17. How many ways can 6 oranges be chosen from a fruit bowl that contains 10 oranges?
Since the order does not matter, the problem is a combination. Therefore the answer
will be:
C10,6 =
10!
= 210
6!(10 - 6)!
18. In a lottery game where 4 numbers are chosen from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9},
how many different 4-number cards could be made where:
a) order is taken into consideration?
P10,4 =
10!
= 5040
(10 - 4)!
b) order is not taken into consideration?
C10,4 =
10!
= 210
4!(10 - 4)!
19. In a lottery game where 5 numbers are chosen from the set {10, 11, 12, . . . , 28, 29} how
many different 5-number cards could be where:
a) order is taken into consideration?
P20,5 =
20!
= 1,860, 480
(20 - 5)!
b) order is not taken into consideration?
C20,5 =
20!
= 15,504
5!(20 - 5)!
7-15
Copyright © 2015 Pearson Education, Inc.
Finite Mathematics
Chapter 7
20. From a standard deck of cards (52 cards) how many 6-card hands will contain:
a) Four jacks and two queens?
C4,4 ×C4,2 =
4!
4!
×
4!(4 - 4)! 2!(4 - 2 )!
= 1 ×6
=6
b) Three jacks and three queens?
C4,3 ×C4,3 =
4!
4!
×
3!(4 - 3)! 3!(4 - 3)!
= 4 ×4
= 16
c) Two jacks and four queens?
C4,2 ×C4,4 =
4!
4!
×
2!(4 - 2)! 4!(4 - 4 )!
= 6 ×1
=6
d) How many 6-card hands will contain only jacks and queens?
6 +16 + 6 = 28
7-16
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