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# Discrete Structures 2

A branch of mathematics that deals with separable and distinct numbers with graph and logical statements are included, and numbers can be finite or infinite.

## propositional logic

Which of the following graphs is not a characteristics of isomorphic graphs?

• Graphs with the same number of vertices.
• Graphs with the same adjacency matrices.
• Graphs with different adjacency matrices.
• Graphs with the same number of edges.

Are the two graphs isomorphic?

• Cannot be determined
• No
• Yes

How many leaves does it have?

• 9
• 8
• 4
• 6
• 2

What is the negation of a tautology?

• Binary Negation
• Unary Negation
• Implication

What is the coefficient of x101 y99 in the expansion of (2x-3y)200?

• C(200,99) (2)101 (-3)99
• C(200,99) (2)99 (-3)101
• C(200,101) (2)99 (-3)101
• C(200,101) (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?

• k is prime, k is not prime
• k = 3n, k ≠ 3n
• the product ends in 3, the product ends in 6, the product ends in 9
• 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

What is the coefficient of aby98 in the binomial expansion of (ab+y)99?

• 98
• 99
• 100
• 4,851

This is a statement that is always false.

• implication
• tautology
• conjunction

Which logical operator represents the statement “if and only if”?

• implication
• conditional
• conjunction
• biconditional

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?

• 2k = + 2k+1 - 1
• 2k+1 - 1
• 1 + 2 + 22 + 23 + 24 + ... + 2k = 2k+1 - 1
• 1 + 2 + 22 + 23 + 24 + ... + 2k + 2k+1

It is impossible for a valid argument to have a true premise and

• a false conclusion
• a true conclusion
• a negated conclusion
• a conditional conclusion

The number of leaf nodes in a complete binary tree of depth d is

• 2d+1
• 2d+1+1
• 2d
• 2d-1+1

Consider the following finite automaton A over Σ = {a,b,c}:

• ɛ ∈ L(A)
• bbaacbabcac ∈ L(A)
• bacabca ∈ L(A)
• The automaton A is a Deterministic Finite Automaton (DFA).

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?

• Neither A nor B
• Both A and B
• A only
• B only

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)

• C(26,3) · C(10,3)
• C(26,3) · 103
• 263 · 103
• 26 · 25 · 24 · 23 · 103

How many terms does the binomial expansion of (2x+3)99 has?

• 101
• 99
• infinite
• 100

What is the probability that a person in the survey selected at random favors using cameras to identify red-light runners?

• 0.31
• 0.64
• 0.59
• 0.69

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?

• inclusion-exclusion
• product rule
• pigeonhole principle
• sum rule

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.

• 8.57

Refer to the graphs below:

• The graphs are not isometric
• The graphs are isometric.
• The graphs are both complete and not isomorphic.
• The answer cannot be determined with the given graphs.

Which is not an invariant when determining if graphs are isomorphic?

• same degree of vertices
• existence of closed loops
• same number of edges
• same number of vertices

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?

• C(12,4)
• 412
• 12! · 4!
• 124

Suppose inflation decreases the value of money by 3% per year? Which formula describes an = the value (in dollars) of \$1000 after n years?

• an = -0.03 · 1000
• an = n (0.97)n 1000
• an = (0.97)n 1000
• an = 1000 (0.03)n

The following breakdown of a total of 18,686 transportation fatalities that occured in 2007 was obtained from records compiled by the U.S. Department of Transportation (DOT). Mode of Transportation Car Train Bicycle Plane Number of Fatalities 16,525 842 698 538 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.

• 0.07
• 0.11
• 0.05
• 0.08

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"

• B only
• Neither A nor B
• A only
• Both A and B

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.

• an = 1.03(an-1 + 1000)
• an = 1.03an-1 + 1000
• none of the given
• an = (1000 + 0.03)an-1

Every tree with at least two nodes has at least two nodes of what degree?

• 3
• 0
• 1
• 2

How many nonisomorphic simple graphs are there with five vertices and three edges?

• 2
• 4
• 1
• 3

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.

• 0.35

What type of graph is depicted below?

• Simple Graph
• Isograph
• Pseudograph
• Directed Graph

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?

• The probability is 0.01
• The probability is 0.04
• The probability is 0.02
• The probability is 0.03

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?

• C(12,3) + C(10,3)
• C(12,3) · C(10,3)
• C(22,6)
• C(22,6) - C(12,6) - C(10,6)

Which of the following is not a recurrence relation?

• primality of numbers
• geometric sequence
• simple and compound interest computations
• fibonacci sequence

How many edges does a tree with V vertices have?

• V - 1
• infinite
• V2
• V
• V + 1
• 0

What is the 8th term in the binomial expansion of (2x+y)16?

• 5,857,280 x8 y8
• 5,857,280 x16 y16
• 5,857,280 x7 y9
• 5,857,280 x9 y7

How many of the nodes have at least one sibling?

• 8
• 7
• 9
• 5
• 6

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)

• 62 - 526
• 62 - 266
• 62 - 366
• 62 - 106

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.

• {(1, 2), (5, 3), (6, 6)}
• {(3, 1), (2, 6), (4, 4)}
• {(1, 2), (2, 4), (3, 6)}
• {(1, 1), (2, 1), (1, 6)}

Which of the following statements about binary trees is NOT true?

• Every binary tree has at least one node.
• Every non-empty tree has exactly one root node.
• Every non-root node has exactly one parent.
• Every node has at most two children.

Which of is a formula for the sequence 3, 6, 12, 24, 48, ...? Assume that the first term in the sequence is called a0?

• none of the given
• an = 3n + 3
• an = 2an-1
• an = 3(n+1)
• an = 3 · 2n-1

A full binary tree with 2n+1 nodes contain

• n non-leaf nodes
• n leaf nodes
• n-1 leaf nodes
• n-1 non leaf nodes

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?

• 40! / (12! · 8! · 20!)
• 20! · 12! · 8!
• C(12,12) · C(8,8) · C(20,20)
• 40! / 3

"A simple graph with 15 vertices with each having a degree of 5 can exist." This statement is ________.

• True
• False

Are the events F and G complementary?

• yes
• no

The preorder and post order traversal of a Binary Tree generates the same output. The tree can have maximum

• Two nodes
• Three nodes
• One node
• Any number of nodes

In order to get the contents of a binary search tree in ascending order, one has to traverse it in.

• not possible
• pre-order
• in-order
• post-order

Finite automata requires minimum _______ number of stacks.

• 3
• 2
• 0
• 1
• 0 (zero)

Two finite state machines are said to be equivalent if they

• recognize same set of tokens
• have same number of states and edges
• have same number of states
• have same number of edges

Logic is a system based on __________.

• statements
• propositions
• truth values
• truth tables

The basic limitation of finite automata is that

• All of the mentioned
• It cannot process strings of length greater than 5
• It sometimes recognize grammar that are not regular.
• It sometimes fails to recognize regular grammar.
• It cannot remember arbitrary large amount of information.

A Finite State Automaton can have more than one initial state.

• True
• False

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?

• 24 ≥ 42
• 22 ≥ 22
• 23 ≥ 32
• 20 ≥ 02
• 21 ≥ 12

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?

• If n is not divisible by 30 then n is divisible by 2 or divisible by 3 or divisible by 5.
• If n is not divisible by 30 then n is not divisible by 2 or not divisible by 3 or not divisible by 5.
• If n is divisible by 2 and divisible by 3 and divisible by 5 then n is divisible by 30.
• If n is not divisible by 2 or not divisible by 3 or not divisible by 5 then n is not divisible by 30.

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)?

• C(52,7) - C(40,7)
• C(4,1) · C(4,1) · C(4,1) · C(40,4)
• C(12,3) · C(40,4)
• C(12,1) · C(40,6)

Translate the expression to predicate logic: “No students are allowed to carry guns.”

• c

Suppose you wish to prove this statement "If n is an integer, then n ≤ n3." Which of the following is correct?

• The given statement is true and can be proven easily using a direct proof
• The given statement is true and can be proven easily using mathematical induction
• The given statement is false because a counterexample can be found.
• The given statement is true and can be proven easily using contradiction

Which concept considers the arrangement of objects / elements in a set?

• permutations
• counting theory
• pigeonhole
• combination

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?

• 1 + 2 + 22 + 23 + ... + 2k = 2k+1 - 1 -> 1 + 2 + 22 + 23 + ... + 2k + 2k+1 = 2k+2 - 1
• 1 + 2 + 22 + 23 + ... + 2k = 2k+1 - 1 -> 1 + 2 + 22 + 23 + 24 + ... + 2k + 2k+1 - 1
• 1 + 2 + 22 + 23 + ... + 2k = 2k+1 - 1 -> 1 + 2 + 22 + 23 + ... + 2k + 2k+1 = 2k+1 - 1 + 2k+1
• 2k -> 2k+1 - 1

How may isomorphic graphs are there for a graph with n number vertices?

• n! number of vertices
• (2n - 1)! number of vertices
• (n + 1)! number of vertices
• (n - 1)! number of vertices

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

• {(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)}
• {(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6)}
• {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}

How many nodes in a tree have no ancestors?

• 2
• 3
• 1
• 0

What is the maximum possible number of nodes in a binary tree at level 6

• 128
• 32
• 64
• 16

The number of different directed trees with 3 nodes is

• 3
• 2
• 4
• 5

A terminal node in a binary tree is called _______________.

• Leaf
• Root
• Bark
• Child
• Branch

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?

• The probability is 0.50
• The probability is 0.40
• The probability is 0.45
• The probability is 0.35

When inorder traversing a tree resulted e a c k f h d b g; the preorder traversal would return.

• abcdefghk
• faekcdhgb
• faekcdbhg
• eafkhdcbg

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.

• 100
• 98
• 198
• 100 x 98
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