SDC Discrete Mathematics I

What kind of reasoning does the statement below illustrate? My phone starts to ring whenever my cat starts to purr.

False cause

Which type of logical fallacy is illustrated below? This chocolate pie must be bad because nobody is taking it.

An appeal to popularity is a type of logical fallacy that says that something is true or false based on popular opinion.

Suppose you have a circular arrangement of three items. If the circle is free, in how many ways can the items be arranged?

1

If point A is found at (1,3) and is translated 6 units up, then the new coordinates would be:

(1,3) + (0,6) = (1,9)

What is the binomial probability formula?

B(x; n, P) = C * P^x * (1 - P)^(n - x)

Which method is efficient? A. Brute force B. Nearest neighbor C. Repeated nearest neighbor D. Cheapest link

B, C, and D.

Which method is optimal for finding the most efficient Hamilton circuit?

Brute force.

Which statement is an example of circular reasoning?

Circular reasoning is when a statement is used to prove itself, such as, 'The moon is made of cheese because the moon is made of cheese.'

Adding an additional edge to a spanning tree would produce _____.

Adding one more edge to a spanning tree produces a loop, which is the same as a cycle. The new edge is redundant. (All are correct.)

What is the difference between a directed and an undirected graph?

In a directed graph, we use arrows on edges to indicate that the relationship indicated by the edge only goes from one vertex to the other, not the other way around. In an undirected graph, the relationship represented by an edge goes both ways between any pair of vertices with an edge between them.

Which of these is the first step in mathematical induction?

Prove the statement is true for the first element in the set.

The converse of a logical statement is found by doing what?

Switching the hypothesis and the conclusion.

Calculate the probability of getting a single pair in a hand of poker. Round that probability to the nearest percent. Please note that a hand is 5 cards.

The total number of ways to get a single pair is given by 13 choose 1 times 4 choose 2 times 12 choose 3 times (4 choose 1)^3, since we're choosing from 13 possible values and 4 suits for the pair of cards and from 12 possible values and 4 possible suits (3 times) for the remaining three cards. This gives 1,098,240 possible ways to get a pair. Divide this by 52 choose 5 (the total number of hands) for a probability of about 42%.

Which rule represents the nth term in the sequence 9, 16, 23, 30...?

an = 7n + 2

The number of comparisons required to sort n items into a tree structure varies as _____.

nlog(n)

An example of a recurrence relation is _____.

un = 3 un-1

If U = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} and A = {12, 13, 14, 20}, what is the complement of A?

{11, 15, 16, 17, 18, 19}

The shape ABC is located at (1,1), (4,0), and (-1,-1). Using matrices, rotate this shape 90 degrees in the counterclockwise direction. What are the coordinates for A'B'C?

(-1,1), (0,4), (1,-1)

When using a Karnaugh Map to simplify an expression, what do you look for? A: Groupings of the 1 output either in a straight line or forming a square or rectangle B: Random locations of the 1 output

Just A

A matrix B multiplied by its identity matrix is equal to what?

Matrix B

Which formula is equivalent to an OR gate?

NOR(NOR(X,Y),NOR(X,Y))

Find the solution to the recurrence relation f(n) = 16 f(n/4) + n.

O(n2)

What is a logic gate?

One or more transistor switches used to convert a binary input into a binary output based on a logical operation

A binomial experiment must have two possible outcomes: success and failure. Select the answer that is an example of a binomial experiment.

Picking an ace (success) or not an ace (failure) out of a deck of cards. Replacing each of the cards before drawing again. Drawing 20 cards total.

Which of the following is an isomorphism?

Plugging in several different values for x and y, the only equation that gives you the same result on both sides is the tan x = sin x / cos x. This is an isomorphism, a change that doesn't change the original function.

A search which starts at the root of the tree and bears left is the _____ strategy.

Preorder

Finding the minimum spanning tree by starting at a random node and adding the node with the lowest weight link is called _____.

Prim's Algorithm

Which of the following is the induction step in mathematical induction?

Show that if the statement is true for the first k elements, then it is true for the (k+1)st element in the set.

A roulette wheel has 38 slots labeled with the numbers 1 through 36 and then 0 and 00. Slots 1 through 36 are colored either red or black. There are 18 red and 18 black. Slots 0 and 00 are colored green (see picture). On one spin of the roulette wheel, what is the probability that the ball lands on a red slot?

Since there are 38 slots, and 18 of them are red, the probability of the ball landing on a red slot is 18/38 which simplifies to 9/19.

What is the 83rd term of the sequence 91, 87, 83, 79, ... ( = a1, a2, a3, a4, ...)?

Since we don't have a0, we'll use the formula with an = d(n - 1) + a1. This gives: an = -4(n - 1) + 91. Plugging in 83: a83 = -4(83 - 1) + 91 = -4(82) + 91 = -328 + 91 = -237.

Say you're playing Blackjack, and you have a four and a jack. If you see that one of the dealer's two cards is a five, then what's the probability that you will be dealt a card that helps you?

Since you want to reach a total of 21 without going over, and you already have a total of 14, the cards that will help you have values between 1 and 7. That means the remaining cards that can help you are the four aces, the four twos, the four threes, the three fours, the three fives, the four sixes, and the four sevens. That's a total of 26 cards out of the remaining 48, so the probability that you will be dealt a card that helps you is: 26/48 = 0.54167, or approximately 54%.

If a universal set is {1, 2, 3, 4, 5, 6, 7} and set C equals {1, 2, 3}, What is the complement of the complement of C?

The complement of C equals {4, 5, 6, 7}. The complement of {4, 5, 6, 7} equals {1, 2, 3}. In other words, we return to the elements of C.

A vehicle license plate uses three numbers and three letters on each plate. The numbers are listed first and then the letters. The numbers used range from 0-9 and the letters used can be any letter of the 26 letters of the alphabet. On any given license plate, the letters can be repeated, but the numbers cannot be repeated. How many different plates are possible?

The correct answer is 10 * 9 * 8 * 26 * 26 * 26 = 12,654,720. There are 10 choices for the first number. However, there are only 9 choices for the second number and 8 choices for the third number because numbers cannot repeat. The letters can repeat, so there are 26 choices for each of the three letters on the license plate. If you chose 17,576,000, then you calculated 10 choices for each number. If you chose 11,232,000, then you did not allow the letters to repeat.

What is the difference between homogeneous linear recurrences and non-homogeneous linear recurrences?

The homogeneous linear recurrences express its elements exclusively as a function of its preceding elements, while the non-homogeneous linear recurrences may also express its elements as a function of the position i of the element.

How do we find the number of items in neither of two sets?

The items in neither of two sets are all of the items outside the union of the two sets. This can be calculated by subtracting the number of items in the union of the two sets from the total number of items in the universe.

When finding the number of items in either of two sets, why does it not work to simply add the number of items in each of the two individual sets?

The items that are in common with both sets will be counted too many times.

The name one-to-one describes which function?

The name one-to-one refers to injective functions.

What does the solution to a recurrence for a divide-and-conquer algorithm estimate?

The number of operations needed to solve the problem.

A given homogeneous linear recurrence has the following characteristic equation: s + 1 = 0. Which is the order of this homogeneous linear recurrence?

The order of a homogeneous linear recurrence can be known according to the highest term, which in this case is s power 1. Therefore, this homogeneous linear recurrence has order 1.

Consider an example with 100 students where 20 are taking discrete math, 30 are taking Java, 25 are taking web design, 6 are taking discrete math and Java, 8 are taking discrete math and web design, 10 are taking Java and web design and 5 are taking all three classes. How many students are taking discrete math only?

First take the number of students taking discrete math and subtract the number of students taking both discrete and Java. Then subtract the number of students taking both discrete and web design: 20 - 6 - 8 = 6. This will subtract the number of students taking all three classes twice, so add the number of students taking all three classes (5) to get 11.

The Boolean NOT operator:

Flips the state (1 or 0) it applies to.

If he eats a hamburger, then he will eat two bags of fries. Which of the following represents the hypothesis in the above conditional statement?

HE EATS A HAMBURGER.

A game of Blackjack is played. Suppose you are the first player dealt with a new pack of cards. What is the probability you get an ace and a jack (in any order)?

The probability of getting an ace is 4/52. Once you get the ace there are 51 cards in the pack. That means the probability of getting a jack is 4/51. Thus the probability of getting an ace first and then a jack is (4/52) x (4/51) = 4/663. However, we do not care about the order, which means we can also look at the probability of getting the jack first and then the ace. Since these probabilities are equal, the probability of getting an ace and then a jack or vice versa is 8/663.

You play a game where you toss a coin. On each toss if it lands with heads up, you win $1. However, if it lands with tails up, you lose $2. If you continue to play this game, how much can you expect to win or lose per game?

The probability of heads is 1/2 and this is worth +$1 to you. The probability of tails is 1/2 and this is worth -$2 to you. So we multiply the value times the probability and then add: (1/2 x 1) + (1/2 x -2) = 0.5 - 1 = -0.5. So you expect to lose $0.50 per game.

What is the ratio of the desired outcome and the total number of possible outcomes?

The theoretical probability is the ratio of the desired outcome and the total number of possible outcomes. If you have the probability of a certain outcome without actually doing the experiment first, then you are working with theoretical probability.

Jeanette asks 50 students in her math class if they are comfortable using technology. Thirty-eight say they are comfortable using technology. Calculate the theoretical and actual probability of this scenario.

Theoretical: 50% Actual: 76%

Which of the following functions is NOT an injection going from the set of real numbers to the set of real numbers?

f(x) = x2

In Mr. Martin's math class there are 15 students out of which 4 will be selected to receive a special award. How many combinations of students can be selected?

n = number of students = 15 r is the number of items being chosen at a time = 4 Number of combination = 15C4 = 15!/4!(15-4!)=1365

A class of 6 boys and 12 girls are preparing for finals. The teacher is going to select 3 students to be exempt from the test. What is the probability that the three students that are exempt will be all boys?

n, is the number of item, and in this case total number of boys and girls = 6+12=18 r, is the number of item being chosen at a time, r = 3 Number of outcomes = 18C3 = 18!/3!(18-3)! =816 Now since there are 6 boys and selection is for 3, n term would be 6 and r term would be 3. Our equation would be 6C3 = 6!/3!(6-3)! = 20 Required probability = 20/816 = 5/204

Which graph in discrete mathematics has a path of edges between every pair of vertices in the graph?

A connected graph

A spanning tree must connect nodes _____.

A spanning tree must connect nodes in no particular order as long as it uses a minimum number of edges.

The set of all elements that are under consideration for a particular problem or situation is known as:

A universal set.

A complete graph is a graph where each vertex is connected to how many of the other vertices? Half None Some All 2

All

Which of these questions should you ask yourself to help determine if mathematical induction is a good method to prove a given statement? Am I trying to prove something is true for an infinite set of elements? If the statement is true for the first k elements, can we use that to show it is true for the (k+1)st element? All of these are questions that could be asked when determining if mathematical induction is a good method of proof to use to prove a statement. Can I prove the first few cases easily?

All questions

What is the binomial probability formula?

B(x; n, P) = n C x * Px * (1 - P)(n - x)

What is the number of successes in a binomial experiment called?

Binomial random variable.

When one event affects the outcome of another event, we call them:

Dependent events

You conduct an experiment where you want to measure the number of rolls it takes to get two 6's in a row when you roll a fair six-sided die. State whether the random variable is discrete or continuous and give a summary of its values.

Discrete with values 2, 3, 4, 5, 6, etc.

If 55 and 89 are two sequential terms of the Fibonacci sequence, what term would come next?

Each term in a Fibonacci sequence is found by adding the two previous terms, so in this case, the next term is 55 + 89 = 144.

Which rule represents the nth term in the sequence 9, 16, 23, 30...?

Each term increases by 7 each time and a1 = 9. a0 = 9 - 7 = 2. an = 7n + 2. Written another way: an = 7(n - 1) + 9.

Which one of the following pairs of sets is equal? A = {3, 5, 7} B = {9, 11, 13} A = {0, 1, 2, 3} |A| = 4 A = {1, 2, 3} B = {3, 1, 2} A = {0, 1, 2, 3} n(A) = 4 A = {1, 2, 3} B = {a, b, c}

Equal sets have the same elements in any order. {1, 2, 3} is therefore equal to {3, 1, 2} and to {3, 2, 1}.

Finding the minimum spanning tree by listing all possible spanning trees, totaling their weights, and selecting the tree with the lowest total weight is called _____.

Explicit enumeration

The value at time n is the value at time (n - 1) plus 6. If the start value at time n = 0 is 4, the value at time n = 2 is _____.

Expressed as a recurrence relation: un = un - 1 + 6. Thus, u1 = uo + 6 = 4 + 6 = 10. And, u2 = u1 + 6 = 10 + 6 = 16.

The total number of degrees in a graph is 20. How many edges does it have?

10

Solve the expression 7P2 (P = permutation)

7 x 6 = 42

Which of the following makes a recursive function?

The function calls itself.

Calculate the number of combinations of royal flushes.

A royal flush is a hand with the highest cards all of the same suit. There are only four combinations of these in one deck.

What do we call the probability of each success in a given binomial experiment?

Binomial distribution

What do we call the probability of each success in a given binomial experiment?

Binomial distribution is the probability of each success in a given binomial experiment.

What is the number of successes in a binomial experiment called?

Binomial random variable

Which of these is NOT true about Boolean logic?

It evaluates whether integers are TRUE or FALSE.

Which of the following is a linear recurrence relation?

un = 3 un - 1

What is the formula for binomial combinations?

n C x = n!/{(n - x)!x!}

How many different combinations of flushes are there in a fair deck of cards?

A flush is five cards of the same suit, but not a royal flush or a straight flush. There are 5,108 combinations of these in one deck.

What's the difference between a sequence and a series?

A sequence is an ordered list of numbers, and a series is the sum of a sequence's terms.

A graph with an Euler circuit can have at most how many odd vertices?

0

How many vertices can have an odd number of edges connecting it to others in a Eulerian circuit?

0

What is the domain of √(x)?

0 to infinity

Solve the expression 5P4 (P = permutation )

120

Suppose you have a drawer full of white socks and black socks. What is the minimum number of socks you would need to pull out of the drawer to guarantee a pair of matching socks?

3

When you multiply matrices, if the first matrix is a 4x3 matrix, which of the following matrices can we multiply with the first?

3x5

Solve the expression 7P2 (P = permutation)

42

Jimmy is making multi-flavored ice cream cones by scooping in different flavors one at a time. Jimmy has 6 different flavors but can only put 3 flavors in each cone. The order of the flavors is important to him as it affects how he tastes each ice cream. How many different arrangements of cones can Jimmy make?

6P3 = 6!/(6 - 3)! = 720/6 = 120 or 6 x 5 x 4 = 120.

You play a game with two six-sided dice. If you roll a sum of 6 or 8, you win $3. If you roll a sum of 11, you win $1, but for anything else, you lose $2. If you continue to play this game, what do you expect to win in the long run?

There are 5 ways to get a sum of 6 when you roll two six-sided dice: 1 and 5, 2 and 4, 3 and 3, 5 and1, 4 and 2. There are also 5 ways to get a sum of 8 when you roll two six-sided dice: 2 and 6, 3 and 5, 4 and 4, 6 and 2, 5 and 3. So the probability of getting a sum of 6 or 8 is 10/36. There are 2 ways to get a sum of 11 when rolling two six-sided dice: 5 and 6, 6 and 5. So the probability of getting a sum of 11 is 2/36. The probability of anything else is 1 - (2/36) - (10/36) = 24/36. The expected value is calculated by multiplying the value of each event by its probability and then summing the results, so we get an expected value of: (3 x 10/36) + (1 x 2/36) + (-2 x 24/36) = -16/36, which is about -0.44.

A locker combination contains four numbers between 1 and 20 and none of the numbers can be repeated. What is the probability that the locker combination will consist of all even numbers?

This problem involves finding two permutations: the number of total possible combinations and the number of possible combinations with only even numbers. The final probability is (even # combos/all possible combos), which reduces to the correct answer. The probabilities for [even numbers/all possible combinations] with no repetition gives us: (10 x 9 x 8 x 7) /(20 x 19 x 18 x 17) When you multiply these together and reduce the product you get 14/323.

Twenty students compete in a school-wide marathon and each student is of comparable running ability. Of the 20 students, 15 were boys and 5 were girls. What is the probability that girls will place 1st, 2nd, and 3rd in the marathon?

To solve this problem, you must look at the probabilities that the top three runners are girls and multiply them together, 5/20 x 4/19 x 3/18 = 1/114.

Let the universal set U be all the letters of the English alphabet. What is the complement of the empty set? (Note: the empty set is a subset of every set.)

U

What is the term for the collection of elements from set A plus the elements in set B?

Union is the set of all elements in either set (or both). The intersection is the set of all elements in common with both sets. The difference is the set of all elements in one set but not in the other set. The complement is the set of all elements not in a given set.

A sandwich shop offers 3 types of meat, 3 types of cheeses, 3 types of bread, and 4 types of condiments for its sandwiches. If you select one of each of the four ingredients, how many possible sandwiches can you create?

Use the fundamental counting principle: the number of sandwiches that can be created is 3 * 3 * 3 * 4 = 108.

If x = 1 and y = 2, what can be said about z if z = xy?

Using logic in math is about mixing the specific language used in logic with the specific symbols used in math. If x = 1 and y = 2, and z = xy, then z = 2.

Children born after the turn of the century have a 60% probability of needing braces. What are the expected value and standard deviation for a group of 30 children surveyed?

We are going to use two different formulas to solve this problem. First, let's figure the expected value. The formula for expected value, or the mean, of a binomial random variable is n * p. Expected value: n * P = 30 * .60 = 18 Next, let's calculate the standard deviation, sqrt(n * P * ( 1 - P )). Standard deviation: sqrt (n*P (1-P)) = sqrt (30 *.6 (1-.6)) = sqrt (18 (.4)) = sqrt (7.2) = 2.7

Ally is trying to determine if an experiment is binomial. She is asking twenty people in front of a pet store if dogs' mouths are cleaner than a toilet. She asks the participants to answer true or false to her question. As far as she knows, the responses of the previous participants do not influence the responses of the other participants. Could this be considered binomial?

We know that Ally's experiment is binomial because she has two outcomes (true or false), the outcomes are independent (she believes the participants do not know the responses of other participants) and there is a fixed number of trials (20 people questioned).

Suppose we wanted to use mathematical induction to prove that for each natural number n, 1 + 5 + 9 + ... + (4n - 3) = n(2n - 1). What would we show in the base step for n = 1 and n = 2? We would show that if the statement is true for the first k elements, then it is true for the (k + 1)st element. We would show that the statement was true for n = 1 and for n = 2 by plugging 1 and 2 into our formula separately, and making sure they both make a true statement. We would show that 2 - 1 = 1. None of these are correct.

We would show that the statement was true for n = 1 and for n = 2 by plugging 1 and 2 into our formula separately, and making sure they both make a true statement.

Discrete math deals with problems concerning options that can be expressed as _____.

Whole numbers

Discrete math deals with problems concerning options that can be expressed as _____.

Whole numbers.

Use De Morgan's Law to rewrite the negation of the statement: You get home by 10pm or you're grounded.

You don't get home by 10PM and you're not grounded