Prelims:
It is impossible for a valid argument to have a true
premise and
Answer: a false conclusion
Translate the following statement into a logical
expression:
You are to use the following predicates:
M(x): x is a MATH233 student
L(x): x loves art
“Every MATH233 student loves art.”
Answer:
Translate the expression to predicate logic:
“No students are allowed to carry guns.”
Answer:
Answer: False
Which symbol is used as the existential quantifier?
Answer:
Translate the expression to predicate logic:
“Some problems are difficult.”
Answer:
Translate the following statement into a logical
expression:
You are to use the following predicates:
M(x): x is a MATH233 student
L(x): x loves art
“Some MATH233 students love art.”
Answer:
Answer: False
Which symbol is used as the universal quantifier?
Answer:
Translate the following statement into a logical
expression:
You are to use the following predicates:
M(x): x is a MATH233 student
L(x): x loves art
“Some students love art.”
Answer:
Answer: True
Let p, q, and r be the following propositions:
P: You get an A on the final exam
Q: You do every exercise in the book.
R: You get an A in this class.
Write the following formula using p, q, and r and
logical connectives.
“Getting an A on the final and doing every exercise
in the book is sufficient for getting an A in this
class.”
Answer:
Suppose that PNo is the statement “n + 1 = n + 2”.
What is wrong with the following proof that the
statement PNo is true for all non negative integers
n?
You assume that P(k) is true for some positive
integer k, that is, that k+1 = k+2. Then you add 1 to
both sides of this equation to obtain k+2 = k+3;
therefore P(k+1) is true. By the principle of
mathematical induction PNo is true for all nonnegative integers n.
Answer: The proof is incorrect because there is no
basis step.
Which of is a formula for the sequence 3, 6, 12, 24,
48, …? Assume that the first term in the sequence is
called a0?
Answer: none of the given
What is the coefficient of x¹⁰¹ y⁹⁹ in the expansion of
(2x-3y)²⁰⁰?
Answer: C(200,99) (2)¹⁰¹ (-3)⁹⁹
Suppose you want to prove that every product of
integers of the form k(k+1)(k+2) is divisible by 6. If
you want to prove this by cases, which of the
following is a set of cases you would use?
Answer: when k is divided by 3, the remainder is 0;
when k is divided by 3, the remainder is 1; when k
is divided by 3, the remainder is 2
Suppose you wish to prove this statement “If n is an
integer, then n ≤ n³.” Which of the following is
correct?
Answer: The given statement is false because a
counterexample can be found.
Suppose you want to use the principle of
mathematical induction to prove that 1 + 2 + 2² + 2³
+ 2³ + … + 2n = + 2n+1 – 1 for all non-negative
integers n. Which of theses is the correct statement
P(k) in the inductive step?
Answer: 1 + 2 + 2² + 2³ + 2⁴ + … + 2k = 2k+1 – 1
What is the 8th term in the binomial expansion of
(2x+y)¹⁶?
Answer: 5,857,280 x⁹ y⁷
Which rule states that if (k + 1) or more objects are
placed into k boxes, then there is at least one box
containing two or more of the objects?
Answer: pigeonhole principle
Suppose inflation decreases the value of money by
3% per year? Which formula describes an = the
value (in dollars) of $1000 after n years?
Answer: an = (0.97)n 1000
Consider the statement, “If n is divisible by 30 then
n is divisible by 2 and by 3 and by 5.” Which of the
following statements is equivalent to this
statement?
Answer: If n is not divisible by 2 or not divisible by
3 or not divisible by 5 then n is not divisible by 30.
What is the negation of a tautology?
Answer: Contradiction
Translate the following statement into a logical
expression:
You are to use the following predicates:
M(x): x is a MATH233 student
L(x): x loves art
“No student loves art.”
Answer:
Which of the proposition is
is:
Answer: Logically equivalent to
Which concept considers the arrangement of
objects / elements in a set?
Answer: permutations
Which logical operator represents the statement “if
and only if”?
Answer: biconditional
Let p and q be propositions.
does not imply
is:
Answer: True
This is a statement that is always false.
Answer: contradiction
Translate the following statement into a logical
expression:
You are to use the following predicates:
M(x): x is a MATH233 student
L(x): x loves art
“Some students are enrolled in MATH233.”
Answer:
Let p, q, and r be the following propositions:
P: You get an A on the final exam
Q: You do every exercise in the book.
R: You get an A in this class.
Write the following formula using p, q, and r and
logical connectives.
“You get an A in this class, but you do not do every
exercise in the book.”
Answer:
Which of the following is a recurrence relation for
the sequence that begins 3, 6, 9, 12, 15, ...?
Answer: an = an-2 + 6
Logic is a system based on __________.
Answer: propositions
Which of the following is not a recurrence relation?
Answer: primality of numbers
Let p, q, and r be the following propositions:
P: You get an A on the final exam
Q: You do every exercise in the book.
R: You get an A in this class.
Write the following formula using p, q, and r and
logical connectives.
“To get an A in this class, it is necessary for you to
get an A on the final.”
Answer:
Midterms:
What type of graph is depicted below?
Answer: Directed Graph
A dormitory has 40 students: 12 sophomores, 8
juniors, and 20 seniors. Which of the following is
equal to the number of ways to put all 40 in a row
for a picture, with all 12 sophomores on the left, all
8 juniors in the middle, and all 20 seniors on the
right?
Answer: 20! · 12! · 8!
Consider the statement “If the product of two
integers is even, then their sum is also even.” Which
of the following assertions is correct?
Answer: The statement is false as you can find a
counterexample.
A class consists of 12 women and 10 men. How
many ways are there to form a committee of size six
if the committee has equal numbers of women and
men?
Answer: C(12,3) · C(10,3)
You have 12 balls, numbered 1 through 12, which
you want to place into 4 boxes, numbered 1
through 4. If boxes can remain empty, in how many
ways can the 12 balls be disturbed among the 4
boxes?
Answer: 4¹²
What is the coefficient of aby⁹⁸ in the binomial
expansion of (ab+y)⁹⁹?
Answer: 99
How many different choices of winners can you
have if the draw is limited to first year and second
year students and you only have one grand prize?
Note: Given that the population of the school is as
follows: 1st year = 100 students, 2nd year = 98
students, 3rd year = 102 students, 4th year = 50
students.
Answer: 198
Let PNo be the statement “you can make n cents
postage using 3-cent and 5-cent stamps.” Suppose
you want to use the Principle of Mathematical
Induction to show that PNo is true for all n ≥ 8.
You begin by proving P(8), which is true because 8
cents postage can be made with one 3-cent stamp
and one 5-cent stamp.
Which of the following will show that the
implication P(k) -> P(k+1) in the inductive step is
true for all k ≥ 8?
Answer: none of the given
Assume that you have an ordinary deck of 52
playing cards. How many possible 7-card poker
hands are there that contain at least one face card
(J, Q, K)?
Answer: C(52,7) – C(40,7)
What type of graph is depicted below?
Answer: Pseudograph
What type of graph is depicted below?
Answer: Simple Graph
How many different passwords are possible if each
password consists of six characters where each
character is either an uppercase letter, a lowercase
letter, or a digit, and at least one digit must be
included in the password? (Note: There are 26
letters and 10 digits)
Answer: 62 – 52⁶
How many terms does the binomial expansion of
(2x+3)⁹⁹ has?
Answer: 100
How many different license plates are available if
the license plate pattern consists of four letters that
cannot be repeated and followed by three digits
that can be repeated? (Assume that all letters are
uppercase and the digits are 0, 1, … 9)
Answer: 26 · 25 · 24 · 23 · 10³
If you give a proof by mathematical induction of the
statement that 2n &ge= n², for all integers n ≥ 4, the
basis step requires you to prove that which of the
following is true?
Answer: 2⁴ ≥ 4²
How many bit strings of length four do not have two
consecutive 1s?
Answer: 8
“A simple graph with 15 vertices with each having a
degree of 5 can exist.” This statement is ________.
Answer: False
Suppose you are hired by a company at an initial
salary of $30,000. At the end of each year you
receive a 3% raise, plus an addition of $1000 on
your base salary. Let an equal your salary at the end
of n years with the company. Find a recurrence
relation for an.
Answer: An = 1.03an-1 + 1000
Suppose you want to use the principle of
mathematical induction to prove that 1 + 2 + 22 +
23 + 23 + … + 2n = + 2n+1 – 1 for all positive integers
n. Which of these is the correct implication p(k) ->
P(k+1) to be used in the inductive step?
Answer: 1 + 2 + 2² + 2³ + … + 2k = 2k+1 – 1 -> 1 + 2 +
2² + 2³ + … + 2k + 2k+1 = 2k+2 – 1
Let S = {x, y, z} be a sample space of an experiment
with outcomes x, y, and z. List all the events of this
experiment.
Answer:
An experiment consists of casting a pair of dice and
observing the number that falls uppermost on each
die. We may represent each outcome of the
experiment by an ordered pair of numbers, the first
representing the number that appears uppermost
on the first die and the second representing the
number that appears uppermost on the second die.
Consider the sample space
Determine the event that the number that falls
uppermost on the first die is greater than the
number that falls uppermost on the second die.
Answer: {(2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (3, 2), (4,
2), (5, 2), (6, 2), (4, 3), (5, 3), (6, 3), (5, 4), (6, 4), (6,
5)}
An opinion poll is conducted among a state’s
electorate to determine the relationship between
their income levels and their stands on a
proposition aimed at reducing state income taxes.
Voters are classified as belonging to either the low-,
middle-, or upper-income group. They are asked
whether they favor, oppose, or are undecided
about the proposition. Let the letters L, M, and U
represent the low-, middle-, and upper-income
groups, respectively, and let the letters f, o, and u
represent the responses – favor, oppose, and
undecided, respectively.
Describe the event E that a respondent does not
favor the proposition and does not belong to the
upper-income group.
Answer:
Are the two graphs isomorphic?
Answer: Yes
Refer to the graphs below:
Answer: The graphs are isometric.
As part of a quality-control procedure, an inspector
at Bristol Farms randomly selects ten eggs from
each consignment of eggs he receives and records
the number of broken eggs.
Describe the event F that at least nine eggs are
broken.
Answer:
Refer to the graphs below:
Answer: The graphs are isometric.
Which is not an invariant when determining if
graphs are isomorphic?
Answer: existence of closed loops
Let
.
Are the events F and G complementary?
Answer: no
An experiment consists of casting a pair of dice and
observing the number that falls uppermost on each
die. We may represent each outcome of the
experiment by an ordered pair of numbers, the first
representing the number that appears uppermost
on the first die and the second representing the
number that appears uppermost on the second die.
Consider the sample space
Determine the event that the sum of the numbers
falling uppermost is greater than or equal to 7.
Answer: {(1, 6), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4,
3), (4, 5), (4, 6), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6,
1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
A study of deaths in car crashes from 1986 to 2002
revealed the following data on deaths in crashes by
day of the week.
Find the empirical probability distribution
associated with these data.
Day of Sunday Monday Tuesday Wednesday
the
Week
Average
131
97
94
99
Number
of
Deaths
Day of the Thursday
Friday
Saturday
Week
Average
103
135
157
Number of
Deaths
Answer: Day of the Week Sunday Monday
Tuesday Wednesday Probability
0.161 0.119
0.115 0.121 Day of the Week Thursday Friday
Saturday Probability
0.126 0.165 0.192
The number of cars entering a tunnel leading to an
airport in a major city over a period of 200 peak
hours was observed and the following data were
obtained.
What is the probability that more than 600 cars will
enter the airport tunnel during a peak hour? Round
to three decimal places, if necessary, and be sure to
express answer in decimal form, not as a
percentage.
The probability is _____.
Answer: 0.6
An experiment consists of selecting a card from a
standard deck of playing cards and noting whether
the card is black (B) or red (R). What are the events
of this experiment?
Answer:
One light bulb is selected at random from a lot of
110 light bulbs, of which 2% are defective. What is
the probability that the light bulb selected is
defective?
Answer: The probability is 0.02
How many nonisomorphic simple graphs are there
with five vertices and three edges?
Answer: 4
A certain airport hotel operates a shuttle bus
service between the hotel and the airport. The
maximum capacity of a bus is 20 passengers. On
alternate trips of the shuttle bus over a period of 1
wk, the hotel manager kept a record of the number
of passengers arriving at the hotel in each bus.
Describe the event E that a shuttle bus carried
fewer than twelve passengers.
Answer:
The number of subscribers to five leading e-mail
services is shown in the accompanying table:
Company
A
Subscribers
320,000
B
C
D
E
210,000 130,000 90,000 50,000
Find the empirical probability distribution
associated with these data.
Answer: Company
A
B
C
D
E
Subscribers 0.4 0.2625 0.1625 0.1125
0.0625
How may isomorphic graphs are there for a graph
with n number vertices?
Answer: n! number of vertices
A sample of two transistors taken from a local
electronics store was examined to determine
whether the transistors were defective (d) or
nondefective (n). What is an appropriate sample
space for this experiment?
Answer:
Refer to the graphs below:
Answer: The graphs are isometric.
An experiment consists of casting a pair of dice and
observing the number that falls uppermost on each
die. We may represent each outcome of the
experiment by an ordered pair of numbers, the first
representing the number that appears uppermost
on the first die and the second representing the
number that appears uppermost on the second die.
Consider the sample space
Determine the event that the number falling
uppermost on one die is a 4 and the number falling
uppermost on the other die is less than 4.
Answer
Strongl Somewhat Somewha Strongl Don
y
support
t
y
kno
support
oppose Oppose
Respondent
410
212
95
s
Answer: {(1, 4), (2, 4), (3, 4), (4, 3), (4, 2), (4, 1)}
164
In a survey of 900 likely voters, the following
question was asked: Do you support using cameras
to identify red-light runners? The results of the
survey follow:
What is the probability that a person in the survey
selected at random favors using cameras to identify
red-light runners?
Answer: 0.69
A pair of fair dice is cast. What is the probability
that the sum of the numbers shown uppermost is
less than 3?
Answer: The probability is 1/36
19
What is the probability of arriving at a traffic light
when it is red if the red signal is flashed for 35 sec,
the yellow signal for 5 sec, and the green signal for
60 sec? Round to two decimal places, if necessary,
and be sure to express answer in decimal form, not
as a percentage.
The probability is _____.
Answer: 0.35
Refer to the graphs below:
Answer: The graphs are not isometric
An experiment consists of selecting a letter at
random from the letters in the word MINNESOTA
and observing the outcomes. What is an
appropriate sample space for this experiment?
Answer:
Refer to the graphs below:
Answer: The graphs are isometric.
The number of cars entering a tunnel leading to an
airport in a major city over a period of 200 peak
hours was observed and the following data were
obtained:
Find the empirical probability distribution for this
experiment.
Answer:
An experiment consists of tossing a coin, rolling a
die, and observing the outcomes. Describe the
event “A head is tossed and an odd number is
rolled.”
Answer:
If a ball is selected at random from an urn
containing four red balls, two white balls, and two
blue balls, what is the probability that it will be a
white ball? Round to two decimal places, if
necessary, and be sure to express answer in decimal
form, not as a percentage.
The probability is _____.
Answer: 0.25
The results of a time study conducted by the
production manager of Ace Novelty are shown in
the accompanying table, where the number of
space action-figures produced each quarter hour
during an 8-hour workday has been tabulated. Find
the empirical probability distribution associated
with this experiment.
Figures
Produced (in
dozens)
30
31
32
33
34
Frequency of
Occurrence
4
0
6
6
8
35
36
Answer: Figures
Produced
6
2
Frequency of
Occurrence
(in dozens) 30
0.1875 33
0.1875 36
0.125 31
0.1875 34
0.0625
0 32
0.25 35
Let
. Find the event
Answer:
An experiment consists of casting a pair of dice and
observing the number that falls uppermost on each
die. We may represent each outcome of the
experiment by an ordered pair of numbers, the first
representing the number that appears uppermost
on the first die and the second representing the
number that appears uppermost on the second die.
Consider the sample space
Determine the event that the number that falls
uppermost on the second die is double that of the
number that falls on the first die.
Answer: {(1, 2), (2, 4), (3, 6)}
A study conducted by the Corrections Department
of a certain state revealed that 163,767 people out
of a total adult population of 1,799,738 were under
correctional supervision (on probation, parole, or in
jail). What is the probability that a person selected
at random from the adult population in that state is
under correctional supervision?
Answer: 0.091
Which of the following graphs is not a
characteristics of isomorphic graphs?
Answer: Graphs with different adjacency matrices.
What is the probability of arriving at a traffic light
when it is red if the red signal is flashed for 30 sec,
the yellow signal for 5 sec, and the green signal for
40 sec?
Answer: The probability is 0.40
An experiment consists of selecting a card from a
standard deck of playing cards and noting whether
the card is black (B) or red (R). Describe an
appropriate sample space for this experiment.
Answer:
One light bulb is selected at random from a lot of
110 light bulbs, of which 5% are defective.
What is the probability that the light bulb selected
is defective? Round to two decimal places, if
necessary, and be sure to express answer in decimal
form, not as a percentage.
The probability is _____.
Answer: 0.05
Are the two graphs isomorphic?
Answer: Yes
The following breakdown of a total of 18,686
transportation fatalities that occurred in 2007 was
obtained from records compiled by the U.S.
Department of Transportation (DOT).
Mode of
Car
Train Bicycle Plane
Transportation
Number of
16,525
842
698
538
Fatalities
What is the probability that a victim randomly
selected from this list of transportation fatalities for
2007 died in a train or a plane accident? Round
answer to two decimal places.
Answer: 0.07
A time study was conducted by the production
manager of Vista Vision to determine the length of
time (t) in minutes required by an assembly worker
to complete a certain task during the assembly of its
Pulsar color television sets.
Describe a sample space corresponding to this time
study.
Answer:
Finals:
How many nodes in a tree have no ancestors?
Answer: 1
The number of different directed trees with 3 nodes
is
Answer: 3
A full binary tree with 2n+1 nodes contain
Answer: n non-leaf nodes
A binary tree of depth “d” is an almost complete
binary tree if.
A. Each leaf in the tree is either at level “d” or at
level “d– 1”
B. For any node “n” in the tree with a right
descendant at level “d” all the left descendants of
“n” that are leaves, are also at level “d”
Answer: Both A and B
When inorder traversing a tree resulted e a c k f h d
b g; the preorder traversal would return.
Answer: faekcdhgb
In order to get the contents of a binary search tree
in ascending order, one has to traverse it in.
Answer: in-order
What is the maximum possible number of nodes in
a binary tree at level 6
Answer: 64
The preorder and post order traversal of a Binary
Tree generates the same output. The tree can have
maximum
Answer: One node
The number of leaf nodes in a complete binary tree
of depth d is
Answer: 2d
Two finite state machines are said to be equivalent
if they
Answer: have same number of states and edges
Consider the following finite automaton A over Σ =
{a,b,c}:
Which of the following statements about A is/are
correct?
Answer: bbaacbabcac ∈ L(A)
Answer: No
The basic limitation of finite automata is that
Answer: It cannot remember arbitrary large
amount of information.
Which of the following statements about binary
trees is NOT true?
Answer: Every binary tree has at least one node.
If two finite states machine M and N are
isomorphic, then
A. M can be transformed to N, merely re-labelling
its states
B. M can be transformed to N, merely re-labelling
its edges
Which is true?
Answer: A only
How many leaves does it have?
Answer: 4
How many edges does a tree with V vertices have?
Answer: V – 1
Transition function maps.
Answer:
Every tree with at least two nodes has at least two
nodes of what degree?
Answer: 1
Finite automata requires minimum _______
number of stacks.
Answer: 0
A terminal node in a binary tree is called
_______________.
Answer: Leaf
How many of the nodes have at least one sibling?
Answer: 6
A Finite State Automaton can have more than one
initial state.
Answer: False