MGF 1106
Review # 1
Prof. Nicoli-Sueo
-----------------Name
This is just a review. You need to work on all your homework assignments and this review to be well prepared for your
exam. They complement each other.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Determine whether the sentence is a statement.
1) Does Chantelle always act like that when she's unhappy?
1)
Given that p and q each represents a simple statement, write the indicated compound statement in its symbolic form.
2) p: This is a herring.
q: This is a fish.
If this is not a herring, then this is not a fish.
2)
3) p: The cone has three scoops.
q: The cone costs $1.85.
The cone has three scoops if and only if the cone costs $1.85.
3)
4) p: She drives at 80 mph.
q: She gets a speeding ticket.
She drives at 80 mph or she does not get a speeding ticket.
4)
Given that p and q each represents a simple statement, write the indicated symbolic statement in words.
5) p: Roger likes Vanessa
q: Vanessa likes Roger
-(p/\q)
5)
6) p: Bob respects Linda
q: Linda respects Bob
-p/\-q
6)
TRUE/F ALSE. Write 'T' if the statement is true and 'F' if the statement is false.
Use the definition of conjunction, /\, to determine the truth value for the statement.
7) Bombay is a city and Brazil is a country.
7)
8) 4 + 3 = 7 and 4 is an odd number.
8)
9) 7 x 2
= 14 or Spanish
is a language.
9)
1
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Complete the truth table by filling in the required columns.
10)
10) - P V - q
p
T
T
F
F
-ql-pV-q
ql-p
T
F
T
F
Construct a truth table for the statement.
11)
11) -(-(q V p»
Use the De Morgan law that states
- -
-(p 1\ q) is equivalent to p V q
to write an equivalent English statement for the statement.
12)
12) It is not true that oaks and dandelions are both trees.
Construct a truth table for the statement.
13) (q
~
p) 1\
-q
13)
-
Use the result that states
- --
p
q is equivalent to q
P
to write an equivalent English statement for the given statement.
14) 1£we are going to a restaurant tonight, then I do not eat lunch.
14)
Construct a truth table for the given statement and then determine if the statement is a tautology.
15) [(p 1\- q) 1\- P ]
16) (
- P V q)
~
(p
~
~
-q
15)
16)
q)
Construct a truth table for the given statement.
17) [ (p 1\ q) 1\ (p
~
q) ]
~
17)
(p V q)
Construct a truth table for the statement. Then determine if the statement is a tautology.
18)
18) (p~q)~(-q~-p)
Write the equivalent to the conditional statement.
19) 1£Michael passes the course, then he graduates from college.
Use a truth table to show that p
to the statement shown.
-
q and
- p V q are equivalent.
20) 1£the garden is not watered every day, the flowers wilt.
2
19)
Then use the result to write a statement that is equivalent
20)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Select the statement that is equivalent to the given statement.
21)
21) I ate a hot dog or a candy bar.
A) If I did not eat a hot dog, then I ate a candy bar.
B) If I ate a hot dog, then I ate a candy bar.
C) If I ate a hot dog, then I did not eat a candy bar.
D) If I did not a hot dog, then I did not eat a candy bar.
Select the statement that is NOT equivalent to the given statement.
22)
22) It is not true that Melanie and Ingrid are both butchers.
A) Melanie is not a butcher and Ingrid is not a butcher.
B) Melanie is not a butcher or Ingrid is not a butcher.
C) If Melanie is a butcher, then Ingrid is not a butcher.
D) If Ingrid is a butcher, then Melanie is not a butcher.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Write the equivalent contrapositive of the statement.
23) If the TV is broken, then I cannot watch my programs.
23)
24) If he is not working in China, then he is vacationing in England.
24)
Write the nonequivalent
converse and inverse of the statement.
25) If you are getting a haircut, then you are not dancing.
25)
26) If it is spring, then some people go hiking.
26)
Write the negation of the conditional statement.
27) If it is red, then it is not an egg.
27)
28) If she can't feed the dog, I will.
28)
29) If there is an earthquake, then all policemen are on call.
29)
Use De Morgan's laws to write a negation of the statement.
30) Roger or Emil will attend the game.
30)
31) Cats are lazy or dogs aren't friendly.
31)
3
un
-~
32) She is not younger than 24 and he is younger than 24.
32)
Translate the argument into symbolic form. Then use a truth table to determine whether the argument is valid or invalid.
(Ignore differences in past, present, and future tense.)
33)
33) There must be a cease-fire or there is fighting.
There is fighting
------------------------
:. There is no cease-fire.
34) If every student passes the quiz, then no review sessions are needed.
Some students do not pass the quiz.
34)
------------------------
:. Some review sessions are needed.
Use a truth table to determine whether the symbolic form of the argument is valid or invalid.
35)P q
~P
35)
---:.
36)
~
q
(p"'" q) A
(q
36)
p)
q
----
:. p V q
Let U = {l, 2, 4, 5, a, b, c, d, e}.Use the roster method to write the complement of the set.
37) A = {2,4, b, d}
37)
Let U = {2I, 22, 23, ..., 40},A = {2I, 22, 23, 24, 25},B = {26,27, 28, 29},C = {2I, 23, 25, 27, ..., 39},and D = {22,24, 26, 28, ..., 40}.
Use the roster method to write the following set.
38) A'
38)
Let U = {2I, 22, 23, 24, ...},A = {2I, 22, 23, 24, ..., 40},B = {2I, 22, 23, 24, ..., 50},C = {22,24, 26, 28, ...},and D = {2I, 23, 25, 27, ...}.
Use the roster method to write the following set.
39) C'
39)
Write!;;; or g in the blank so that the resulting statement is true.
40) {3,5, 7}- {2,3, 4, 5, 7}
40)
Use !;;;,g, c, or both c and!;;; to make a true statement.
41) {a,b}-
{z,a, y, b, x, c}
41)
4
- ---
--
42) {14,15, 16}
-
42)
{14,15, 16}
List all the subsets of the given set.
43) {Siamese, domestic shorthair}
43)
44) {14}
44)
45) (2)
45)
Let U = {q, r, s, t, u, v, w, x, y, z}
A = {q, s, u, w, y}
B = {q, s, y, z}
C = {v, w, x, y, z}. List the elements
in the set.
46) (A n B) u (A n C)
46)
47) A' n (B u C)
47)
48) (A' n B) u (A' n C)
48)
Shade the Venn diagram to represent the set.
49) A' n B'
49)
([)
50) (A n B) u (A u B)'
50)
([)
5
-
- --
__n
n_n
n
__n-
51) (A u B u C)'
51)
Use the union rule to answer the question.
52) If n(A) = 5, n(B) = 11, and n(A n B) = 3; what is n(A u B)?
52)
53) If n(A) = 40, n(B) = 117, and n(A u B) = 137; what is n(A n B)?
53)
54) If n(B) = 36, n(A n B) = 7, and n(A u B) = 63; what is n(A)?
54)
Use a Venn Diagram and the given information to determine the number of elements in the indicated set.
55) n(A) = 55, n(B) = 63, n(C) = 57, n(A n B) = 11, n(A n C) = 13, n(B n C) = 7, n(A n B n C) = 5,
and n(A' n B' n C) = 111. Find n(U)
55)
TRUE/F ALSE. Write 'T' if the statement is true and 'F' if the statement is false.
Use a Venn diagram to decide if the statement is true or false.
56)
56) (A' n B)' = A n B'
(])
57)
57) (A u B')' = A' n B
(])
6
58)
58) (A' u B)' = A' n B
QD
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Use a Venn Diagram and the given information to determine the number of elements in the indicated set.
59) n(U) = 180, n(A) = 60, n(B) = 80, n(A n B) = 25, n(A n C) = 28,n(A n B n C) = 13,
n(A' n B n C) = 42, and n(A' n B' n C) = 45. Find n(C).
59)
60) n(U) = 60, n(A) = 26, n(B) = 16, and n(A n B) = 9. Find n(A u B)'.
60)
Use the union rule to answer the question.
61) If n(A) = 25,n(A u B)= 73, and n(A n B)= 21;what is n(B)?
61)
Use a Venn diagram to answer the question.
62) A local television station sends out questionnaires to determine if viewers would rather see
a documentary, an interview show, or reruns of a game show. There were 800 responses
with the following results:
240 were interested in an interview show and a documentary, but not reruns;
32 were interested in an interview show and reruns, but not a documentary;
112 were interested in reruns but not an interview show;
192 were interested in an interview show but not a documentary;
80 were interested in a documentary and reruns;
48 were interested in an interview show and reruns;
64 were interested in none of the three.
How many are interested in exactly one kind of show?
62)
63) A survey of260 familiesshowed that
99 had a dog;
76 had a cat;
34 had a dog and a cat;
98 had neither a cat nor a dog, and in addition did not have a parakeet;
8 had a cat, a dog, and a parakeet.
How many had a parakeet only?
63)
Write the sample space for the given experiment.
64) A box contains 10 red cards numbered 1 through 10. One card is drawn at random.
64)
65) A box contains 2 blue cards numbered 1 through 2, and 3 green cards numbered 1 through
3. A blue card is picked, followed by a green card.
65)
7
Solve the problem.
66) One card is selected from a deck of cards. Find the probability of selecting a black card or
a queen.
66)
67) One card is selected from a deck of cards. Find the probability of selecting a red card or a
card less than 8. (Note: The ace is considered a low card.)
67)
68) One card is selected from a deck of cards. Find the probability of selecting a red card or a
heart.
68)
69) One card is selected from a deck of cards. Find the probability of selecting a diamond or a
card less than 8. (Note: The ace is considered a low card.)
69)
Suppose
P(C) = .048, P(M n C) = .044, and P(M u C) = .524. Find the indicated probability.
70) P(M)
70)
71) P(M')
71)
72) P(C')
72)
Find the odds in favor of the indicated event.
73) Spinning an A on the spinner pictured below. (The sectors are of equal size.)
73)
74) Spinning a D on the spinner pictured below. (The sectors are of equal size.)
74)
75) Randomly drawing a 5 from the cards pictured below.
75)
[!J[I]0~@]
76) Randomly drawing an even number from the cards pictured below.
[!J[I]0~@]
8
76)
Convert the odds that the given event will occur to the probability that the event will occur.
77) The odds in favor of winning a particular lottery are 1 to 2,000,000.
77)
78) The odds in favor of becoming the President are 1 to 5,700,000.
78)
Solve the problem.
79) Of the coffee makers sold in an appliance store, 3.0% have either a faulty switch or a
defective cord, 2.1% have a faulty switch, and 0.5% have both defects. What is the
probability that a coffee maker will have a defective cord? Express the answer as a
percentage.
79)
80) If a single fair die is rolled, find the probability of a 4 given that the number rolled is odd.
80)
81) If a single fair die is rolled, find the probability of a 5 given that the number rolled is odd.
81)
Use the theoretical probability formula to solve the problem. Express the probability as a fraction reduced to lowest
terms.
82) Use the spinner below to answer the question. Assume that it is equally probable
that the pointer will land on anyone of the five numbered spaces. If the pointer lands
on a borderline, spin again.
82)
Find the probability that the arrow will land on 1 or 2.
You are dealt one card from a 52-card deck. Find the probability that you are not dealt:
83) as.
83)
84) a heart.
84)
You randomly select one card from a 52-card deck. Find the probability of selecting:
85) an ace or a 7?
85)
You are dealt one card from a 52-card deck. Find the probability that you are dealt:
86)
86) a numbered card or a spade
A single die is rolled. Find the odds:
87) in favor of getting a number less than 3.
87)
88) against getting a number less than 4.
88)
9
Solve the problem.
89) A card is dealt from a 52-card deck. What is the probability of not being dealt a queen of
hearts?
89)
90) Rolling a die, what are the odds in favor of getting an even number?
90)
Solve the problem involving probabilities with independent
events.
91) A spinner is used for which it is equally probable that the pointer will land on anyone of
six regions. Three of the regions are colored red, two are colored green, and one is colored
yellow. If the pointer is spun once, find the probability it will land on green and then
yellow.
91)
92) A spinner is used for which it is equally probable that the pointer will land on anyone of
six regions. Three of the regions are colored red, two are colored green, and one is colored
yellow. If the pointer is spun three times, find the probability it will land on green every
time.
92)
Numbered disks are placed in a box and one disk is selected at random.
93) If there are 4 red disks numbered 1 through 4, and 8 yellow disks numbered 5 through 12,
find the probability of selecting a disk numbered 3, given that a red disk is selected.
93)
94) If there are 8 red disks numbered 1 through 8, and 2 yellow disks numbered 9 through 10,
find the probability of selecting a red disk, given that an odd-numbered disk is selected.
94)
95) If there are 6 red disks numbered 1 through 6, and 4 yellow disks numbered 7 through 10,
find the probability of selecting a yellow disk, given that the number selected is less than or
equal to 3 or greater than or equal to 8.
95)
The chart shows the probability of dying from four conditions in the U.s. Express all probabilities as decimals to three
decimal places. Assume all events are mutually exclusive.
Causes of Death
Disease A
Disease B
Disease C
Disease D
Percentage of all Deaths
30.3%
23.0%
5.8%
4.7%
96) What is the probability of dying from disease A or B?
10
96)
Answer Key
Testname: MGFl106-1-REVIEW-FIU
1)
not a statement
2) -
P~
3)P ~
q
-
q
4) P V - q
5)
It is not
true
that
6) Bob and Linda
7)
Roger
likes
Vanessa
and
Vanessa
likes
Roger.
do not respect each other.
TRUE
8) FALSE
9) TRUE
P q -p -q -pV-q
F
T T F F
T
T
10) T F F
F T T F
T
T
F F T T
11) q
T
T
F
F
P
-(-(q V p))
T
F
T
F
12) oaks
T
T
T
F
are not
p q q~ p
TT
T
13)T F T
F T F
F F T
14) If I eat lunch,
trees or dandelions
-
q
we
p q -q pA-q
15)
T
T
F
F
T
F
T
F
F
T
F
T
F
T
F
F
pq-p-pVq
F
F
T
T
(q ~
F
T
F
T
T
16)T
F
F
T
F
T
F
T
F
T
T
p
T
17) T
F
F
q pAq pq
T T
T
F F
F
T F
T
F F
T
p)
A
-
are not
trees.
q
F
T
F
T
are not
-p
F
F
T
T
going
to a restaurant
tonight.
(pA-q)A-p
F
F
F
[(pA-q)A-p]-q
T
T
T
F
T
p~q (-pVq)~(p~q)
T
T
F
T
T
T
T
T
(pAq)A(p--q) pVq
T
T
F
T
F
T
F
F
Is a tautology.
Is a tautology.
[(p A q) A (p -- q) ]
T
F
F
T
11
(p V q)
Answer Key
Testname: MGFII06-1-REVIEW-FIU
q p-+q -q
T T
F
F F
T
T T
F
F T
T
18)
p
T
T
F
F
19)
If Michael
does
-p -q-+-p
F
T
F
F
T
T
T
T
not
graduate
(p -+ q) +-+ (- q -+ - p)
T
T
T
T
from
college, then
Is a tautology.
he does
not pass
the course.
p q p-+q -p -pVq
TT
T
F
T
20) T F F
F
F
F T T
T
T
F F T
T
T
Either
the garden
is watered
every
programs,
then
day,
or the flowers
wilt.
21)A
22)A
23)
If I can
24)
If he is not vacationing
25)
watch
converse:
inverse:
26)
my
If you
If you
converse:
are not
If some
people
If it is not
27)
It is red
and
28)
She
then
then
you
a haircut,
go hiking,
spring,
then
no
are getting
then
then
one
is not broken.
he is working
you
in China.
a haircut.
are dancing
it is spring
goes
hiking.
it is an egg.
can't feed
the dog,
29) There
is an earthquake
30)
Roger
will not
31)
Cats
32)
She
33)
p: There
aren't
dancing,
are not getting
inverse:
the TV
in England,
lazy
is younger
than
I won't.
and
attend
and
and
some
the game
dogs
are
24
he
or
policemen
and
Emil
are not on call.
will not attend
the game.
friendly.
is not
younger
than
24.
is a cease-fire.
q: There is fighting.
pVq
q
---:.- p
p
q p V q (p V q) II q - P
T T
TF
F T
F F
T
T
T
F
T
F
T
F
F
F
T
T
[(p V q) II q] -+ - P
F
T
T
T
Argument
12
is invalid.
Answer Key
Testname: MGFII06-1-REVIEW-FIU
34) p: Every student passes the quiz
q: No review sessions are needed.
p-q
-p
----
:. - q
p qp-q
TT
T
TF
F
FT
T
FF
T
p q p-q
TT
35) T F
FT
FF
p qp
TT
36) T F
FT
FF
T
F
T
T
-
-p(p-q)A-p
F
F
F
F
T
T
T
T
-q[(p-q)A-p]--q
F
T
T
T
F
F
T
T
Argument is invalid.
- p (p - q) A - P - q [ (p - q) !\ - P ] - -q
F
F
T
T
qq
T
F
T
T
F
F
T
T
-
p [(p
T
T
F
T
F
T
F
T
-
q) A (q
T
F
F
T
T
T
F
T
-
p)] p V q [[( p
T
T
T
F
The argument is invalid.
-
q) A (q
T
T
T
T
-
p)] A q]
Argument is valid.
37) {l/ 5/ a, c/ e}
38) A' = {26,27/ 28/
39)
40)
41)
42)
43)
44)
45)
46)
47)
48)
49)
0
0
0
/
40}
C' = {21, 23/ 25/ 27/ ..o}
k
c and k
k
{Siamese, domestic shorthair}, {Siamese}, {domestic shorthair}, { }
{l4}, { }
no subsets
{q, s/ w/ y}
{r, t/ z}
{r, t/ z}
13
-
(p V q)
Answer Key
Testname: MGFII06-1-REVIEW-FIU
50)
51)
52)
53)
54)
55)
56)
57)
58)
59)
60)
61)
62)
63)
64)
13
20
34
260
FALSE
TRUE
FALSE
61
27
69
384
21
{l,2, 3, 4, 5, 6, 7, 8, 9, 10}
65) {(I, 1), (1,2), (1, 3), (2, 1), (2,2), (2, 3)}
66)
.!.13
67) .!Q.
13
68) 1.
2
69) ..!Z.
26
70)
71)
72)
73)
74)
75)
76)
.520
.480
.952
3 to 5
1 to 7
1 to 4
2 to 3
77)
1
2,000,001
14
Answer Key
Testname: MGFII06-1-REVIEW-FIU
1
78) 5,700,001
79) 1.4%
80) 0
81) 13
82) ~5
83)E13
84) l4
85) 213
86) ..!Q.
13
87) 1:2
88) 1:1
89) 2!.
52
90) 1:1
1
91) 18
1
92) 27
93) 14
94) .!5
95) 12
96) 0.533
15