College Mathematics: Learning Worksheets Chapter 7
185
Name ________________________________ Date ______________ Class ____________
Goal: To convert statements in symbolic logic form and create truth tables
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.
2. George will play cards or he will play pool.
3. The cake was not burned and very tasteful.
4. Margaret did not pass physics.
p
5. Hank did not come home for spring break and he had a wonderful time.
p
Section 7-1 Logic
p q
p
p
q
p
q
p
q
T T F T T T
T F F T F F
F T T T F T
F F T F F T
STATEMENT: pq If p then q
CONVERSE: qp If q then p
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6. The committee decided to table the motion and bring it up at the next meeting.
7. Carol decided to take a math class or a psychology class.
8. Nathan did not agree to go to the movies.
9. If Mary and John decide on a restaurant at which to eat dinner, then they can make
reservations.
10. If Peggy does not go to work today, then she will have to take a personal day from
work.
11. If tomorrow is Sunday, then today is Saturday.
Converse: If today is Saturday, then tomorrow is Sunday.
12. If 56 7+=, then 12 3+= .
Converse: If 12 3+=, then 56 7+=.
13. If two lines intersect, then the lines are not parallel.
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.
Solutions:
15. 16.
p q (
p
∨ q)
p q (
p
q
TTT F F T
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17.
18.
19.
p q (
p
q) ∧(qp)
20.
p q
p
q
p q p ∧ ()
p
q
p q
()
pq∨
()
pq∧
T T T T T
T F T F F
College Mathematics: Learning Worksheets Chapter 7
Name ________________________________ Date ______________ Class ____________
Goal: To find the unions, intersections, and complements of sets using Venn diagrams and
sets in roster form.
In Problems 1–10 write the resulting set using the listing method.
3. {, ,, ,} {,,, , }abcde aeiou 4. {, ,, ,, , , ,} {,,, ,}abcde f ghi aeiou∪
5. {1, 2,3, 4,5,6,7,8,9} {prime numbers < 10}
6. {2,4,6,8,10,12,14,16} {prime numbers < 10}∪
7. For {,,, ,}Uaeiouand {, },Aio=find .
A
8. For {1,2,3,4,5,6,7,8,9,10}Uand {2,4,6,8,10},A=find .A
Section 7-2 Sets
Definitions: Union {| or }
A
BxxAxB
Intersection: { | and }
A
BxxA xB
Complement: {|}
A
xUx A
9. For {prime number < 25}U=and {2,5,13,17},A= find .
A
10. For {positive integers < 15}Uand {positive integers divisible by 3 and < 15},A
find .A
12. If {1, 2},T=list all the subsets of T.
13. If {1, 2 , 3} ,M=list all the subsets of M.
College Mathematics: Learning Worksheets Chapter 7
191
By referring to the Venn Diagram in Problems 14–16, find how many elements are in the
indicated sets.
14.
a) 80 20 50 150 300
15.
a) 25 15 30 30 100
b) 0
16.
a) 21 3 46 48 118
b) 0
c) 3
College Mathematics: Learning Worksheets Chapter 7
193
Name ________________________________ Date ______________ Class ____________
Goal: To solve problems using the basic counting principles
1. a) Determine the number of ways that the elements of the set
{
}
,,abc can be
arranged by listing the possible outcomes.
b) Determine the number of ways that the elements of the set
{
}
,,abc can be
arranged by using a tree diagram.
c
b
b
a
c) Determine the number of ways that the elements of the set
{
}
,,abc can be
arranged by using the multiplication principle.
Section 7-3 Basic Counting Principles
Theorem: Addition Principle (for counting)
For any two sets: ( ) () () ( )nA B nA nB nA B
194
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 fired, green, or yellow).
a) Determine the number of ways that Carol can have the gift wrapped by using a
tree diagram.
R
Y
b) Determine the number of ways that Carol can have the gift wrapped by using the
multiplication principle.
3. Joe and Fred go out to dinner at a local restaurant. On the menu are listed five different
salads and ten 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?
b) How many choices does Fred have if he chooses both a salad and an entrée?
4. Natalie and Maria go shopping for blouses and capris. Natalie has just received her
paycheck and plans to buy both a blouse and a pair of capris. Maria has to make her car
payment out of her paycheck and can only afford to buy a blouse, or capris. The store
they like to shop in has seven styles of blouses and nine styles of capris. How many
choices does Natalie have? How many choices does Maria have?
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?
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?
College Mathematics: Learning Worksheets Chapter 7
196
College Mathematics: Learning Worksheets Chapter 7
197
Name ________________________________ Date ______________ Class ____________
Goal: To solve problems using permutations and combinations
In Problems 1–6, evaluate the permutation or combination.
1. 8,2
P 2.
7,4
P 3.
10,4
P
4. 8,2
C 5.
7,4
C 6.
10,4
C
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.
8. The number of ways letters can be assigned for a car license plate.
9. The number of ways students can be chosen to serve on a committee.
Section 7-4 Permutations and Combinations
Permutation: ,(1)( 2)( 1)
nr
Pnnn nr L or
,
!
!
nr
n
Pnr
11. The number of ways a 5-card hand can be dealt from a standard deck of cards.
12. The number of ways nine students can register for a class.
13. In a mathematics club, eight 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?
14. How many ways can 10 people line up in a cafeteria line?
15. How many ways can seven people line up in a cafeteria line?
16. How many ways can five people be chosen from a group of 13 to attend a conference?
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17. How many ways can three oranges be chosen from a fruit bowl that contains 11 oranges?
18. In a lottery game where 3 numbers are chosen from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9},
how many different 3-number cards could be made where
a) order is taken into consideration?
b) order is not taken into consideration?
19. In a lottery game where 4 numbers are chosen from the set {10, 11, 12, . . . , 28, 29} how
many different 4-number cards could be where
a) order is taken into consideration?
b) order is not taken into consideration?
200
20. From a standard deck of cards (52 cards) how many 5-card hands will contain
a) Four jacks and one queen?
b) Three jacks and two queens?
c) Two jacks and three queens?
d) One jack and four queens?
e) How many 5-card hands will contain only jacks and queens?