Discrete Math Final Exam

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Identify the form of the given argument and determine whether or not it is valid. No even number is divisible by 3. 18 is an even number. :. 18 is not divisible by 3

Valid; Modus Ponens

True or false: The graphs G and G' shown below are isomorphic. Isomorphic Graphs 5.PNG

False These are not isomorphic The graph G is not connected, but G' is connected.

True or false: The graphs H and H' shown below are isomorphic. Isomorphic Graphs 14.PNG

False These are not isomorphic. H has a vertex of degree 4, but H' does not.

Four married couples have reserved eight seats in a row at the theater, starting at an aisle seat. In how many ways can they arrange themselves if all the women sit together and all the men sit together?

1152 There are 4! = 24 ways to seat the men and 4! = 24 ways to seat the women. And there are two ways to arrange the "blocks" of men/women: the four men can be seated on the aisle (with the women inside) or vice versa. So by the Fundamental Principle of Counting, there are 24 x 24 x 2 = 1152 arrangements

How many ways can a president, vice-president, and secretary be chosen from a club with 12 members?

1320

Given: n(A U B UC) = 149 n(A ∩ B ∩ C) = 20 n(A ∩ B) = 42 n(A ∩ C) = 39 n(B ∩ C) = 37 n(A) = 101 n(B) = 74 n(C) = 72 Find n(A' ∩ B ∩ C). Hint: Use the information above to fill in the Venn diagram:

17 Make a Venn diagram with three circles. Begin filling the number in each of the three regions starting with n(A ∩ B ∩ C) Then use that fill in each of n(A ∩ B) , n(B ∩ C) and n(A ∩ C), etc.

Given: n(U) = 60 n(A) = 27 n(B) = 17 n(A ∩ B) = 2 Find n(A U B)c.

18 Use the formula: n(A U B) = n(A) + n(B) - n(A ∩ B) and subtract the result from n(U)

The library is to be given 5 books as a gift. The books will be selected from a list of 21 distinct titles. If each book selected must have a different title, how many possible selections are there?

20,349 In selecting the books, there will be no repetitions (each has a different title) and order will not matter. That is, we will use be counting combinations. There are 21 questions, and we are choosing 5, so the total number of selections is: 21C5 = 20,349.

How many 5-card poker hands consisting of 2 aces and 3 kings are possible from an ordinary 52-card deck?

24 In selecting the cards, there will be no repetitions and order will not matter. That is, we will use be counting combinations. There are 4 Aces, and we are choosing 2, so the total number of ways to choose the aces is: 4C2 = 6. There are 4 Kings, and we are choosing 3, so the total number of ways to choose the black cards is: 4C3 = 4. Using the fundamental principle of counting, there are 6 x 4 = 24 ways to get a five card hand with 2 Aces and 3 Kings.

A club which consists of 14 men and 12 women needs to choose an executive committee consisting of 5 club members. How many different committees consisting of 3 men and 2 women are possible?

24,024

A club which consists of 14 men and 12 women needs to choose an executive committee consisting of 5 club members. How many different committees consisting of 3 men and 2 women are possible?

24,024 In selecting the members of the committee, there will be no repetitions and order will not matter. That is, we will use be counting combinations. There are 14 men, and we are choosing 3, so the total number of ways to choose the men is: 14C3 = 364. There are 12 women, and we are choosing 2, so the total number of ways to choose the women is: 12C2 = 66. Using the fundamental principle of counting, there are 364 x 66 = 24,024 ways to choose the committee.

Given n(A) = 60 n(B) = 68 n(C) = 62 n(A ∩ B) = 12 n(A ∩ C) = 14 n(B ∩ C) = 8 n(A ∩ B ∩ C) = 6 n(A' ∩ .B' ∩ C') = 121 Find n(U).

283 Note that n(A' ∩ B' ∩ C') = 121 is the total number of elements outside all of the sets. Use the PIE formula: n(A U B U C ) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C) to get the total number of elements that are in at least one set. Add the two values to get n(U) Then, subtract the calculated value from 118. ************************** Alternatively, make a Venn diagram with three circles. Begin filling the number in each of the three regions starting with n(A ∩ B ∩ C) Then use that fill in each of n(A ∩ B) , n(B ∩ C) and n(A ∩ C), etc. The total for all eight regions will be the answer.

Monticello residents were surveyed concerning their preferences for candidates Moore and Allen in an upcoming election. Of the 800 respondents, 300 support neither Moore nor Allen, 100 support both Moore and Allen, and 250 support only Moore. How many residents support Moore or Allen?

500 Use the formula: n(A U B) = n(A) + n(B) - n(A ∩ B)

Given that n(A) = 16, n(B) = 45 and n(A U B) = 53. What is n(A ∩ B)?

8 Use the formula: n(A U B) = n(A) + n(B) - n(A ∩ B)

There are 9 members on a board of directors. If they must form a subcommittee of 6 members, how many different subcommittees are possible?

84 If we are selecting a committee, then there will be no repetitions and order will not matter. That is, we will use be counting combinations. There are 9 members of the board, and we are choosing 6, so the total number of committees is: 9C6 = 84.

Of the 2,598,960 different five-card poker hands possible from a deck of 52 playing cards, how many would contain 2 black cards and 3 red cards?

845,000 In selecting the cards, there will be no repetitions and order will not matter. That is, we will use be counting combinations. There are 26 black cards, and we are choosing 2, so the total number of ways to choose the black cards is: 26C2 = 325. There are 26 black cards, and we are choosing 3, so the total number of ways to choose the black cards is: 26C3 = 2600. Using the fundamental principle of counting, there are 325 x 2600 = 845,000 ways to get a five card hand with 2 black cards and 3 red cards.

Of the 2,598,960 different five-card poker hands possible from a deck of 52 playingcards, how many would contain 2 black cards and 3 red cards?

We wish to find a minimum weight Hamiltonian circuit for the graph: Graph for NNA or CLA6.PNG Apply the nearest neighbor algorithm to the graph starting at vertex A. State the circuit obtained from the algorithm as a list of vertices, starting and ending at vertex A. Example: ABCDEA.

ABDECA

Given the weighted graph below, we will apply Kruskal's algorithm to find a minimal spanning tree. Graph for NNA or CLA10.PNG

AC CD 39

We wish to find a minimum weight Hamiltonian circuit for the graph: Graph for NNA or CLA10.PNG Apply the nearest neighbor algorithm to the graph starting at vertex A. State the circuit obtained from the algorithm as a list of vertices, starting and ending at vertex A. Example: ABCDEA.

ACDBEFA Step 1: Pick the vertex that is closest to A; i.e. select the edge of least weight.Steps 2, 3, 4, 5: At each step, pick the edge of least weight that does not repeat a vertex already visited. Step 6: Return to vertex A

Given the weighted graph below, we will apply Kruskal's algorithm to find a minimal spanning tree. Kruskal Graph 6.PNG

AD AE 26

Given the weighted graph: Graph for NNA or CLA5.PNG We wish to find a minimum weight Hamiltonian circuit starting and ending at vertex A. To do this, we will apply the Cheapest Link Algorithm.

AD DB ADBCEA 34

Given the weighted graph below, we will apply Kruskal's algorithm to find a minimal spanning tree. Kruskal Graph 7.PNG

AE BF 25

Given the statement: If you eat your dinner, then you will get dessert. Answer each of the following:

Answer 1: If you get dessert, then you ate your dinner. Answer 2: If you do not eat your dinner, then you will not get dessert. Answer 3: If you did not get dessert, then you did not eat your dinner. Answer 4: You ate dinner and you did not get dessert.

True or false: The graph K12 has an Euler circuit.

FALSE

Let A, B and C be arbitrary sets, and U a universal set. Let n(A), n(B) represent the cardinality of A, B respectively. For parts a, b and c, determine if the statement is true (i.e. always true) or false. For part d, select the best answer: a) n(A − B) = n(A) − n(B) [ Select ] b) If A U C = B U C, then A = B True c) (A ∩ B)c = Ac U Bc [ Select ] d) Describe the conditions under which the statement A U Ac = U is true.

FALSE FALSE TRUE ALWAYS TRUE

Determine whether the statement is true or false: If G is a connected graph with a Hamiltonian circuit, then every vertex of G has degree 2

False

Determine whether the statement is true or false: If G is a graph with n vertices and n - 1 edges, then G is a tree.

False

Determine whether the statement is true or false: There exists a tree G that has six vertices and total degree 14.

False

True or false: The graph K11,14 has an Euler circuit.

False

True or false: The graphs G and G' shown below are isomorphic. Isomorphic Graphs 11.PNG

False

True or false: The graphs G and G' shown below are isomorphic. Isomorphic Graphs 6.PNG

False

Determine whether the statement is true or false: There exists a connected graph G that has six vertices, five edges, and a non-trivial circuit.

False If connected graph G with six vertices and five edges, must be a tree; hence such a graph would be circuit-free. Thus no such graph exists.

True or false: The graphs G and G' shown below are isomorphic. Isomorphic Graphs 6.PNG

False These are not isomorphic The graph G is not connected, but G' is connected

Given the weighted graph: Weighted K5 no 3.PNG We wish to find a minimum weight Hamiltonian circuit starting and ending at vertex A. To do this, we will apply the Cheapest Link Algorithm.

BE EA AEBDCA 31

Let N represent the natural numbers; i.e. N is the set of non-negative integers. Let A be a finite set with cardinality n > 1. Let P(A) be the power set of A. Define C : P(A) → N be given by: C(B) = |B| I.e. for a subset B of A, C(B) is the cardinality of B. Which of the following is true?

C is neither injective nor surjective. The range of C is {0, 1, 2, 3, . . . , n}; hence C is not surjective. E.g. there is no subset B of A such that |B| = n + 1. And C is not injective, since there are n subsets of cardinality 1. E.g. if A = {a1, a2, . . . , an}, then C({a1}) = C({a2}) = . . . = C({an}) = 1. Thus, C is not one to one.

Given the weighted graph below, we will apply Kruskal's algorithm to find a minimal spanning tree. Graph for NNA or CLA8.PNG

CF DB 83

Let A, B and C be arbitrary sets, and U a universal set. Let n(A), n(B) represent the cardinality of A, B respectively. For parts a, b and c, determine if the statement is true (i.e. always true) or false. For part d, select the best answer: a) n(A U B) = n(A) + n(B) b) if A ⊆ B then A ∩ Bc = ∅. c) A ∩ (B U C) = (A ∩ B) U C d) Describe the conditions under which the statement A U Ac = U is true.

False True False Always true

Select the negation for each of the following statements: a) Fish have scales or spiders spin webs. Fish do not have scales and spiders do not spin webs. b) Jupiter is a star and New Orleans is a city. Jupiter is not a star and New Orleans is not a city. c) Some movies are comedies. [ Select ] d) If it gets cold, then the pipes will freeze.

Fish do not have scales and spiders do not spin webs. Jupiter is not a star or New Orleans is not a city. No movies are comedies. It gets cold, and the pipes do not freeze.

Select the negation for each of the following statements: a) Fish have scales or spiders spin webs. Fish do not have scales and spiders do not spin webs. b) Jupiter is a star and New Orleans is a city. Jupiter is not a star or New Orleans is not a city. c) Some movies are comedies. [ Select ] d) If it gets cold, then the pipes will freeze.

Fish do not have scales and spiders do not spin webs. Jupiter is not a star or New Orleans is not a city. No movies are comedies. It gets cold, and the pipes do not freeze.

Given the statement: If Fluffy is a carnivore, then Fluffy is a dog.

If Fluffy is a dog, then Fluffy is a carnivore. If Fluffy is not a carnivore then Fluffy is not a dog. If Fluffy is not a dog, then Fluffy is not a carnivore. Fluffy is a carnivore but Fluffy is not a dog.

Given the statement: If P is a rectangle, then P is a square. Answer each of the following: a) The converse of the statement above is: [ Select ] b) The inverse of the statement above is: [ Select ] c) The contrapositive of the statement above is: [ Select ] d) The negation of the statement above is:

If P is a square then P is a rectangle. If P is not a rectangle then P is not a square. If P is not a square, then P is not a rectangle. P is a rectangle and P is not a square.

Given the statement: If Sofia passes Algebra, then Sofia can take Finite Math. Answer each of the following:

If Sofia can take Finite Math, then Sofia passed Algebra. If Sofia does not pass Algebra, then Sofia can not take Finite Math. If Sofia cannot take Finite Math, then Sofia did not pass Algebra. Sofia passes Algebra, but Sofia can not take Finite Math.

Identify the form of the given argument and determine whether it is valid or invalid: If it's Tuesday, then this must be Paris. Today is not Tuesday . :. This must not be Paris.

Invalid - Fallacy of the Inverse

Identify the form of the given argument, and determine whether it is valid or invalid. If I'm hungry, then I will eat. I'm not hungry. I will not eat.

Invalid - Fallacy of the Inverse Symbolically, this argument looks like: p → q ~p . :. ~q This is the Fallacy of the Inverse, which we know to be INVALID.

Identify the form of the given argument and determine whether it is valid or invalid: If you read, then you will have a high score. You do not read. You will not have a high score.

Invalid - Fallacy of the inverse

Identify the form of the given argument and determine whether or not it is valid. All businesses are subject to safety inspections. This restaurant is subject to safety inspections. :. This restaurant is a business.

Invalid; Fallacy of the Converse

Select the negation for each of the following statements: a) Napoleon Bonaparte was a general or Florence Nightingale was a nurse. [ Select ] b) Saturn is a planet and Miami is a city. [ Select ] c) Some dogs are terriers. [ Select ] d) If an animal is a chimpanzee, then it is a primate. [ Select ]

Napoleon Bonaparte was not a general and Florence Nightingale was not a nurse. Saturn is a not a planet or Miami is not a city. No dogs are terriers. An animal is a chimpanzee, but it is not a primate.

Define a relation R on the the set Z of all integers by the rule: m R n ⇔ 5 | m2 − n2 For parts a - d, decide if the statement is true or false. For part e, select the best answer: a) The relation R is reflexive. [ Select ] b) The relation R is symmetric. [ Select ] c) The relation R is transitive [ Select ] d) The relation R is anti-symmetric [ Select ] e) How would we best describe the relation R? [ Select ]

TRUE TRUE TRUE FALSE R is an equivalence relation Let R be the on Z given by: m R n ⇔ 5 | m2 − n2 a) R is reflexive Proof: Let m ∈ Z. Then 5 | m2 − m2 (since 5 | 0). Thus, m R m. b) R is symmetric. Proof: Let m, n ∈ Z such that m R n. Then 5 | m2 − n2 I.e. there is a k in Z so that 5k = m2 − n2 Thus, 5(-k) = n2 − m2 and so 5 | n2 − m2 It follows that n R m. c) R is transitive. Proof: Let m, n, p ∈ Z such that m R n and n R p. Then 5 | m2 − n2 and 5 | n2 − p2 I.e. there exist j, k in Z so that 5j = m2 − n2 and 5k = n2 − p2 Thus, 5(j + k) = m2 − n2 + n2 − p2 = m2 − p2 It follows that m R p. d) R is not anti-symmetric, because it is symmetric e) Since R is reflexive, symmetric and transitive, it is an Equivalence Relation

Let A, B and C be arbitrary sets, and U a universal set. Let n(A), n(B) represent the cardinality of A, B respectively. For parts a, b and c, determine if the statement is true (i.e. always true) or false. For part d, select the best answer: a) n(A ∪ B) + n(A ∩ B) = n(A) + n(B) [ Select ] b) If A ⊆ C and B ⊆ C then A ∪ B ⊆ C. [ Select ] c) (A ∩ B)c = Ac ∩ Bc [ Select ] d) Describe the conditions under which the statement A U B = A is true. [ Select ]

TURE TRUE FALSE B ⊆ A

Select the negation for each of the following statements: a) The Bears will win and the Packers will lose. [ Select ] b) The Cubs will win or the Brewers will lose. [ Select ] c) All dogs are carnivores. [ Select ] d) If it snows more than six inches, classes will be cancelled . If it snows more than six inches, classes will not be cancelled.

The Bears will lose or the Packers will win. The Cubs will lose and the Brewers will win. Some dogs are not carnivores. It snows more than six inches, but classes are not cancelled.

Given the recurrence relation: an = 7an−1 + 21an−2; a0 = 6, a1 = 13. Solve the recurrence relation to fill in the blanks:

The characteristic equation is x2 - 4x - 21 = 0 The equation above factors as (x - 7)(x + 3) = 0, so the general solution is: an = c7n + d(-3)n Set a0 = 4 = c + d a1 = 22 = 7c - 3d Solve the system to get c = 4 and d = 2 so the particular solution is:

Given the recurrence relation: an = 9an−1 + 14an−2; a0 = 10, a1 = 25. Solve the recurrence relation to fill in the blanks:

The characteristic equation is x2 - 9x + 14 = 0 The equation above factors as (x - 7)(x - 2) = 0, so the general solution is: an = c7n + d2n Set a0 = 10 = c + d a1 = 25 = 7c - 3d Solve the system to get c = 1 and d = 9 so the particular solution is: an = 7n + 9 . 2n

Consider the graph: Graph for Euler Circuit Problem13.PNG Which of the following is true?