Math 270A Final Review

A={a,b} B={1,2,3} Select the expression that is an element of A×B×B. A) (b,2^2) B) (b,2,3) C) (a,a,1) D) (2,1,1)

(b,2,3)

Use the prefix tree below to encode the word "piece". (Do not add any spaces or special characters to you answer.)

1,110,100,011,110

To select a stir-fry dish, a restaurant customer must select a type of rice, protein, and sauce. There are two types of rices, three proteins, and seven sauces. How many different kinds of stir-fry dishes are available? Note: Give a numerical answer, not an expression with operations

2*3*7 = 42

A state's license plate has 7 characters. Each character can be a capital letter (A-Z), or a digit except for 0 (1-9). How many license plates are there in which exactly 3 of the 7 characters are digits? A) (7 3) ⋅ (26)^4 B) (7 3) ⋅ 9^3 ⋅ (26)^4 C) ( 7 3) ⋅ (35)^4 D) P(7,3)⋅(35)^4

B) (7 3) ⋅ 9^3 ⋅ (26)^4

How many ways are there to permute the letters in SOUPSPOONS? A) 10!/3!2!2! B) 10!/3!3!2! C) 10!/3!3!2!2! D) 10!3!2!

B) 10!/3!3!2!

Select the value of the sum ∑3k=−2k^2 A) 14 B) 19 C) 3 D) 9

B) 19

There is a set of 14 jobs in the printer queue. Two of the jobs in the queue are called job A and job B. How many different ways are there for the jobs to be ordered in the queue so that job A is first or job B is last or both? A) 13! - 12! B) 2 * 13! - 12! C) 13! D) 2 * 13!

B) 2 * 13! - 12!

Select the set of properties such that it is impossible to have a graph that satisfies all the properties in the set. A) 8 vertices, 10 edges, not a tree B) Acyclic, 7 vertices, 7 edges C) Tree, 10 vertices, 9 leaves D) Connected, 7 vertices, 7 edges

B) Acyclic, 7 vertices, 7 edges

Q(n) is a statement parameterized by a positive integer n. The following theorem is proven by induction: Theorem: For any positive integer n, Q(n) is true. What must be proven in the inductive step? A) For any integer k≥1k≥1 , Q(k) implies Q(n). B) For any integer k≥1k≥1 , Q(k) implies Q(k+1). C) For any integer k≥1k≥1 , Q(k). D) For any integer k≥1k≥1 , Q(k-1) implies Q(k).

B) For any integer k≥1k≥1 , Q(k) implies Q(k+1).

Select the graph that has an Euler circuit. A) K3,3 B) K3,4 C) K6 D) K5

B) K3,4

f : Z → Z . f ( x ) = ⌈ x / 3 ⌉ Select the correct description of the function f. A) One-to-one and onto B) Onto but not one-to-one C) One-to-one but not onto D) Neither one-to-one nor onto

B) Onto but not one-to-one

Select the mathematical statements to correctly fill in the beginning of the proof of an inductive step below: We will assume for k≥1k≥1 that 7 evenly divides 6^2k−1 and will prove that 7 evenly divides 6^2(k+1) − 1. Since, by the inductive hypothesis, 7 evenly divides 6^2k−1, then 6^2k can be expressed as (A?), where m is an integer. Now, 6^2(k+1)−1=6^2⋅6^2k−1 = (B?) by the ind. hyp. =... A) (A):7m (B):36(7m)−1 B) (A):7m (B):36(6^2k)−1 C) (A):7m+1 (B):36(7m+1)−1 D) (A):7m+1 (B):36(6^2k)−1

C) (A):7m+1 (B):36(7m+1)−1

A basket holds a set of balls. Each ball is red, green, or blue. How many balls must there be in the basket in order to guarantee that there are at least 5 balls of the same color? A) 15 balls B) 12 balls C) 13 balls D) 14 balls

C) 13 balls

Which statement is false? A) 7 | 0 B) 4 | − 16 C) 2 ∤ 5 D) 1 ∤ 5

C) 2 ∤ 5

How many binary strings of length 10 have at least 2 adjacent bits that are the same ("00" or "11") somewhere in the string? A)(2^10)/2 B) 2^10 C) 2^10−2 D) 2^10−1

C) 2^10−2

The inductive step of an inductive proof shows that for k≥4 , if 2k≥3k , then 2^k+1≥3(k+1). In which step of the proof is the inductive hypothesis used? 2^k+1≥2⋅2^k (Step 1) ≥2⋅3k (Step 2) ≥3k+3k (Step 3) ≥3k+3 (Step 4) ≥3(k+1) (Step 5) A) Step 1 B) Step 4 C) Step 2 D) Step 3

C) Step 2

Which choice corresponds to the level 2 vertices? A) j, f, and i B) g, b, and c C) e, d, m, a, and l D) j, d, k, and i

C) e, d, m, a, and l

Which choice corresponds to the descendants of vertex d? A) d, j, and k B) j and k C) g, b, c, and h D) d, g, b, c, and h

C) g, b, c, and h

Select the value for x that is a counter-example to the following statement: For every integer x , x < x^2 . A) x = -1/2 B) x = -1 C) x = 1 D) x = 1/2

C) x = 1

Select the graph that does not have an Euler trail.

C) { (a,b) (a,c) , (a,d), (a,e) , (b,c), (b,e), (c,d), (d,e) }

A = {a, b, c, d} X = {1, 2, 3, 4} The function f : A → X is defined as f = {(a, 4), (b, 1), (c, 4), (d, 4)} Select the set corresponding to the range of f. A) {1, 2, 3, 4} B) {1} C) {1, 4} D) ∅

C) {1, 4}

The domain for variable x is the set of all integers. Select the statement that is false. A) ∀ x ( x^2 ≥ x ) B) ∀ x ( x^2 ≠ 5 ) C) ∀ x ( x^2 > x ) D) ∃ x ( x = x )

C) ∀ x ( x^2 > x )

Which statement is the contrapositive of: "If x=4, then 3x=12." A) If x = 4 , then 3 x = 12 . B) If x ≠ 4 , then 3 x ≠ 12. C) If 3 x = 12 , then x = 4 . D) If 3 x ≠ 12 , then x ≠ 4 .

D) If 3 x ≠ 12 , then x ≠ 4.

Give a function f mapping vertices from G to G' that is an isomorphism from G to G'. Use proper function notation.

f(a) = 1 f(b) = 4 f(c) = 2 f(d) = 5 f(e) = 3 Or f(a) = 3 f(b) = 5 f(c) = 2 f(d) = 4 f(e) = 1

Use the definition below to select the statement that is false. A={x∈Z:x is even and 4<x<17} A) 6 ∈ A B) | A | = 7 C) 4 ∉ A D) 17 ∉ A

|A| = 7

The domain of discourse are the students in a class. Define the predicates: S(x): x studied for the test A(x): x received an A on the test Select the logical expression that is equivalent to: "Someone who did not study for the test received an A on the test."

∃x(¬S(x)∧A(x))

What is the coefficient of the c^4d^7 term in (−3c+5d)^11? Note: Leave your answer in computation form. You do not need to give a final numerical value.

(11 choose 7) * (-3)^4 * 5^7

A={x∈Z:x is a prime number} B={4,7,9,11,13,14} C={x∈Z:3≤x≤10} Express the set corresponding to (A∪B)∩C(A∪B)∩C in roster notation.

(3, 4, 5, 7, 9}

Natasha is in a class of 30 students that selects 4 leaders. How many ways are there to select the 4 leaders so that Natasha is one of the leaders? (a) 30 4 (b) 29 4 (c) 30 3 (d) 29 3

(d) 29 3

A sequence {a^n} is defined as follows: a^0=2, a^1=1, and for n≥2,a^n=3⋅a^n−1−n⋅a^n−2+1 What is a^3?

-2

f(x) = |x| g(x) = -5x + 2 Compute the correct value for g ∘ f ( 2 )

-8

Use Mathematical Induction to prove that for every positive integer n, 4 evenly divides 32n−1 .

10 pts

Use the prefix tree below to decode 10110011110.

101 + 100 + 1111 + 0 = rice

Graph G is defined by the arrow diagram below. What is the out-degree of vertex 2?

2

What is the common ratio of the following geometric sequence? 2, 6, 18, 54, ...

2

A particular state's license plates have 7 characters. Each character can be a capital letter, or a digit except for 0. How many license plates are there in which no two adjacent characters are the same? Show the computation you would use to get the answer. Do not give the final value.

35 * (34)^6

What is the degree of each vertex of the graph K5 ?

4

What is the height of the tree shown below?

4

A country has two political parties, the Reds and the Blues. The 100-member senate has 44 Reds and 56 Blues. Each party must elect a chair and a vice chair from their party's members, and one person cannot be elected for both. How many different outcomes are there for the chair and vice chair elections? Show the computation you would use to get the answer. Do not give the final value.

44 * 43 * 56 * 55

The first few numbers in the 17th row of Pascal's triangle are: 1, 17, 136, 680, 2380, ... What is the value of (183)(183) ? Explain your answer.

816 18 = n and 3 =k (18,3) = ( (18-1),(3-1)) + ((18-1),3) (17,2)+(17,3) 136 + 680 = 816

Minimum Spanning Tree (MST)

A minimum spanning tree (MST) of a weighted graph, is a spanning tree T of G whose weight is no larger than any other spanning tree of G. (Find the path of least weight on the basis of spanning tree)

The degree sequence of a graph is a list of the degrees of all of the vertices in non-increasing order. The degree sequence for four different graphs are given below. Each graph is guaranteed to be connected. Select the degree sequence corresponding to the graph that has an Euler trail. A) 2, 2, 3, 4, 4, 4, 5 B) 2, 2, 2, 4, 4, 4, 4 C) 2, 4, 4, 4, 4, 4, 6 D) 1, 2, 3, 3, 4, 4, 4, 5

A) 2, 2, 3, 4, 4, 4, 5

How many binary strings of length 10 are there in which the number of 0's in the string is not equal to the number of 1's in the string? A) 2^10−(10 5) B) 2^10 C) 2^9 D) 2^10−2^9

A) 2^10−(10 5)

There is a set of 10 jobs in the printer queue. Two of the jobs in the queue are called job A and job B. How many ways are there for the jobs to be ordered in the queue so that either job A or job B finish last? A) 2⋅9! B) 10! C) 9! D) 10!/2

A) 2⋅9!

Select the mathematical expression that is equivalent to the sum 2+6+18+54+⋯+2⋅3^15 A) 3^16−1 B) (3^16−1)/2 C) 3^13−1 D) (3^15−1)/2

A) 3^16−1

Select the set of properties such that it is possible to have a graph that satisfies all the properties in the set. A) 8 vertices, 7 edges, not a tree B) A free tree, 10 vertices, one leaf C) Connected, 7 vertices, 5 edges D) Acyclic, 6 vertices, 7 edges

A) 8 vertices, 7 edges, not a tree

20 applicants from a pool of 90 applications will be hired. How many ways are there to select the applicants who will be hired? A)(90) (20) B) 20^90 C) P(90,20) D) 90^20

A)(90) (20)

Prove: For any odd integer n , the number 2 n 2 + 5 n + 4 is odd.

Assume: n is an odd integer. so n = 2k +1 for some integer k. then 2n^2 + 5n + 4 = 2(2k +1)^2 + 5(2k +1) + 4 = 2(4k^2+4k+1)+ (10k + 5) + 4 = (8k^2+8k+2)+ (10k + 5) + 4 = 8k^2 + 18k + 11 = 2(4k^2 + 9k + 10) + 1 where 4k^2 + 9k + 10 is an integer therefore 2n^2 + 5n + 4 is odd.

Theorem: For any real number x, if 0 ≤ x ≤ 3 , then 15 − 8 x + x 2 > 0 Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem?

Assumed: 15 − 8 x + x 2 ≤ 0 Proven: x < 0 or x > 3

Theorem: For any two real numbers, x and y, if x and y are both rational then x + y is also rational. Which facts are assumed and which facts are proven in a direct proof of the theorem?

Assumed: x is rational and y is rational Proven: x + y is rational

Select the mistake that is made in the proof given below. Theorem. The product of an even integer and any other integer is even. Proof. Suppose that x is an even integer and y is an arbitrary integer. Since x is even, x = 2k for some integer k. Therefore, xy = 2m for some integer m, which means that xy is even.

Assuming facts that have not yet been proven.

How many strings of length 10 over the alphabet {a, b, c, d} have exactly 3 a's? A) 4^ 7 B) (10 3) ⋅ 3^7 C) (10 3) ⋅ 4^ 7 D) (10 3)

B) (10 3) ⋅ 3^7

How many binary strings of length 12 have exactly six 1's or begin with a 0? A) (12 6)+2^11 B) (12 6)+2^11−(11 6) C) (12 6)+2^11−(11 7) D) (12 6)+2^1

B) (12 6)+2^11−(11 6)

Natasha and Rodrigo are in a class of 30 students that selects 4 leaders. How many ways are there to select the 4 leaders so that either Natasha and Rodrigo are both selected or Natasha and Rodrigo are both not selected? A) (28 2)+(30 4) B) (28 2)+(28 4) C) (30 2)+(30 4) D) (302)+(284)

B) (28 2)+(28 4)

The inductive step of an inductive proof shows that for k≥4. if 2^k≥3, then 2^k+1≥3(k+1). Which step of the proof uses the fact that k≥4≥1 ? 2^k+1≥2⋅2^k (Step 1) ≥2⋅3k (Step 2) ≥3k+3k (Step 3) ≥3k+3 (Step 4) ≥3(k+1) (Step 5) A) Step 5 B) Step 4 C) Step 2 D) Step 3

B) Step 4

Select the words that correctly complete the following sentence: In the graph below, edge {B,C} and vertex B _____ . A) both have degree 3 B) are incident C) are adjacent D) are regular

B) are incident

Which choice corresponds to the leaves of the tree? A)e, g, b, h, m, a, and l B) e, g, b, h, f, m, a, and l C) e, g, b, c, h, f, m, a, and l D) g, b, h, f, m, a, and l

B) e, g, b, h, f, m, a, and l

Select the relation that is an equivalence relation. The domain is the set {1, 2, 3, 4}. A) { (3, 4), (4, 3), (1, 3), (3, 1), (1, 1), (2, 2), (3, 3), (4, 4) } B) { (1, 4), (4, 1), (1, 1), (2, 2), (3, 3), (4, 4) } C) { (1, 4), (4, 1), (2, 2), (3, 3) } D) { (1, 4), (4, 1), (1, 3), (3, 1), (2, 2) }

B) { (1, 4), (4, 1), (1, 1), (2, 2), (3, 3), (4, 4) }

Select the sequence that is a cycle in the graph below: A) ⟨B,D,G,F⟩ B) ⟨B,D,G,E,F,B⟩ C) ⟨B,D,G,E,D,B⟩ D) ⟨B,D,G,F,E,B⟩

B) ⟨B,D,G,E,F,B⟩

Use Mathematical Induction to prove that for every positive integer n, 2+4+6+...+2n=n(n+1)

Base case: n=1 2(1)=?1(1+1) 2=2 check Inductive Step: Assume that 2+4+6+...+2k=k(k+1) for some integer k >=1. Now, 2+4+6+...+2k+2(k+1)=k(k+1)+2(k+1) by our assumption = (k+1)(k+2) which is what we wanted. We have shown that if 2+4+6+...+2n=n(n+1) is true for n=k, then it is true for n=k+1. Therefore, 2+4+6+...+2n=n(n+1) is true for every positive integer n.

Which of the following tic-tac-toe configurations has a child that is a leaf if X plays next? (Check all that apply)

C and D

Select the summation expression whose value is equivalent to the sum 43+63+83+⋯+28343+63+83+⋯+28^3 . A)∑28j=2(2j)^3 B)∑14j=4(2j)^3 C)∑14j=2(2j)^3 D) ∑28j=4(2j)^3

C)∑14j=2(2j)^3

How many strings of length 10 over the alphabet {a, b, c, d} have at least one b somewhere in the string? Note: You do not have to give a final numerical value

Complement rule:# length 10 strings over the alphabet {a,b,c,d} - # length 10 strings over the alphabet {a,c,d} (no b's) = # length 10 strings with at least 1 b 4^10 - 3^10

Prove that the statement "∀A,B,C(A∪C⊆B∪C→A⊆B)" is false.

Counterexample: Suppose A = { 2 }, B = { }, C = { 2, 3 } Then it follows: A U C = { 2, 3 } and B U C = { 2, 3 } so A U C is a subset of B U C But A is not a subset of B ( { 2 } is not a subset of { } ). Therefore, the statement above is false.

10 identical copies of a movie will be stored on 40 computers such that each computer has at most 1 copy. How many different ways can the 10 copies be stored? A) P(40,10) B) 40^10 C) 10^40 D) (40 10)

D) (40 10)

A country has two political parties, the Reds and the Blues. The 100-member senate has 44 Reds and 56 Blues. How many ways are there to pick a 10 member committee of senators with the same number of Reds as Blues? A) (44 5)+(56 5) B) (44 10)⋅(56 10) C) (100 10)(100 10) D) (44 5)⋅(56 5)

D) (44 5)⋅(56 5)

How many strings of length 12 over the alphabet {a, b, c, d} start with "aa" or end with "aa" or both? A) 2⋅(12 2)⋅4^10+(12 4)⋅4^8 B) 2⋅(12 2)⋅4^10−(12 4)⋅4^8 C) 2⋅4^10+4^8 D) 2⋅4^10−4^8

D) 2 * 4^10 - 4^8

A bank PIN is a string of four digits, each digit 0-9. How many choices are there for a PIN if the last digit must be odd? A) 5⋅3^10 B) 10^4 C) 4^10 D) 5⋅10^3

D) 5⋅10^3

A bank PIN is a string of four digits, each digit 0-9. How many choices are there for a PIN if the last digit must be odd and all the digits must be different from each other? A) 5⋅10^3 B) 10⋅9⋅8⋅7 C) 10⋅9⋅8⋅5 D) 9⋅8⋅7⋅5

D) 9⋅8⋅7⋅5

A license plate has 7 characters. Each character can be a capital letter or a digit except for 0. How many license plates are there in which no character appears more than once and the first character is a digit? A) 9⋅(34)^6 B) 9⋅P(35,6) C) 9⋅(35)^6 D) 9⋅P(34,6)

D) 9⋅P(34,6)

The domain of relation R is Z × Z . (a, b) is related to (c, d) if a ≤ c and b ≤ d. Which statement correctly characterizes the relation R? A) R is an equivalence relation. B) R is not an equivalence relation because R is not reflexive.term-18 C) R is not an equivalence relation because R is not transitive. D) R is not an equivalence relation because R is not symmetric.

D) R is not an equivalence relation because R is not symmetric.

What are the edges in the minimum spanning tree of the graph shown below? A) {d, f}, {f, c}, {c, h}, {c, e}, {e, g}, {c, a}, {a, b} B) {d, f}, {d, g}, {g, e}, {e, c}, {c, h}, {f, a}, {a, b} C) {d, f}, {d, g}, {g, e}, {e, c}, {c, h}, {c, a}, {a, b} D) {d, f}, {f, c}, {c, h}, {c, e}, {e, g}, {f, a}, {a, b}

D) {d, f}, {f, c}, {c, h}, {c, e}, {e, g}, {f, a}, {a, b}

In implementing Prim's algorithm, after vertices b, c, e, and f have been added to the Minimal Spanning Tree, what are the set of eligible edges? A) {e, d}, {f, d}, {c, d}, {c, f}, {b, a}, {b, g}, {f, g}, {a, c}, {c, h} B) {e, d}, {f, d}, {c, d}, {c, f}, {b, g}, {f, g}, {a, c}, {c, h}, {a, h} C) {e, d}, {f, d}, {c, d}, {b, g}, {f, g}, {a, c}, {c, h}, {a, h} D) {e, d}, {f, d}, {c, d}, {b, a}, {b, g}, {f, g}, {a, c}, {c, h}

D) {e, d}, {f, d}, {c, d}, {b, a}, {b, g}, {f, g}, {a, c}, {c, h}

A class of 30 students with 14 sophomores and 16 freshmen must select 4 leaders. How many ways are there to select the 4 leaders so that at least one freshman is selected? A) 16⋅(29 3) B) (30 4)−(16 4) C) 16⋅(30 3) D)(30 4)−(14 4)

D)(30 4)−(14 4)

Write the truth table for the proposition p→(q∨¬r).Use insert table along with symbols v, ^, ~, and -> as needed. Caution: The table menu bar will get in your way if you put your table at the very top of the text box. You can get around this by typing some extra text at the top of you text box before inserting the table. Type, "I love math," or your favorite (appropriate) one-liner, etc.

I love math p | q | r | ~r | (q∨¬r) | p→(q∨¬r) ---------------------------------- T | T | T | F | T | T | F | T | T | F | T | F | T | F | F | T | F | T | T | F | F | T | F | T | F | F | T | F | F | F | F | T |

Select the statement that is false. a) If 3 is a prime number, then 5 is a prime number. b) If 4 is a prime number, then 6 is a prime number. c)f 4 is a prime number, then 5 is a prime number. d) If 3 is a prime number, then 6 is a prime number.

If 3 is a prime number, then 6 is a prime number.

Write the contrapositive of: "If x≠4, then 3x≠12."

If 3x = 12, then x=4

Select the expression that is equivalent to the following statement: Among any two consecutive positive integers, there is at least one integer that is not prime.

If x is a positive integer, then x is not prime or x+1 is not prime.

Prove: If m and n are odd integers, then mn+3 is an even integer.

Let m and n be odd integers Then m=2k+1 and n=2j+1 for integers k and j. Now, mn+3 = (2k+1)(2j+1)+3 by substitution = 4jk + 2k +2j + 1 +3 = 4jk + 2k + 2j + 4 by algebra = 2(2jk + k + j + 2) where 2jk+k+j+2 is an integer Therefore, mn+3 is even.

f : Z ^+ → Z ^+ , f ( x ) = x + 3 Select the correct description of the function f. A) Neither one-to-one nor onto B) One-to-one and onto C) Onto but not one-to-one D) One-to-one but not onto

One-to-one but not onto

20 applicants are interviewed for a job. The interviewer creates an ordered list (first to last) of the best 6 applicants. How many ways are there for the interviewer to create the list? Note: You do not need to give a final numerical value

P(20,6)

10 different movies will be stored on 40 computers such that each computer will have at most one movie. Since the movies are different, it matters which movie is stored on which computer. How many different ways can the 10 movies be stored? Note: You do not need to give a final numerical answer.

P(40,10)

Prim's algorithm for finding and MST.

Prim's algorithm finds a minimum spanning tree of the input weighted graph. Input: An undirected, connected, weighted graph G. Output: T, a minimum spanning tree for G. T := ∅. Pick any vertex in G and add it to T. For j = 1 to n-1 Let C be the set of edges with one endpoint inside T and one endpoint outside T. Let e be a minimum weight edge in C. Add e to T. Add the endpoint of e not already in T to T. End-for

The domain of a relation R is the set of integers. x is related to y under relation R if x 2 = y. Is the relation reflexive, symmetric, and/or anti-symmetric?

Reflexive: ["No"] Symmetric: [ "No"] Anti-symmetric: ["Yes"]

Use De Morgan's law to write a statement (in English) that is logically equivalent to: "It is not true that every student got an A on the test."

There is a student who did not get an A on the test.

The domain for variable x is the set {Ann, Ben, Cam, Dave}. The table below gives the values of predicates P and Q for every element in the domain. Name P(x) Q(x) Ann F F Ben T F Cam T T Dave T T Select the statement that is false. a) ∃x (P(x) → Q(x)) b) ∃x (P(x) ∧ ¬Q(x)) c) ∃x (P(x) ∧ Q(x)) d) ∃x (¬P(x) ∧ Q(x))

d) ∃x (¬P(x) ∧ Q(x))

Select the function that does not have a well-defined inverse. A) f : Z → Z f ( x ) = ⌈ x + 2 ⌉ B) f : R → Z f ( x ) = ⌈ x ⌉ C) f : R → R f ( x ) = 3 x + 4 D) f : R → R f ( x ) = − 2 x + 5

f : R → Z f ( x ) = ⌈ x ⌉

A = {a, b, c, d} X = {1, 2, 3, 4} The function f: A → X is defined by the arrow diagram below. Give the set of pairs that defines a function that is equal to f. (For example, f = {(a, 1), (b, 3), (c, 4), (d, 4)})

f = {(a, 2), (b, 3), (c, 4), (d, 2)}

A = {a, b, c, d} X = {1, 2, 3, 4} Each choice defines a function whose domain is A and whose target is X. Select the function that has a well-defined inverse. A) f = {(a, 3), (b, 4), (c, 2), (d, 1)} B) f = {(a, 3), (b, 3), (c, 3), (d, 3)} C) f = {(a, 3), (b, 4), (c, 2), (d, 4)} D) f = {(a, 3), (b, 4), (c, 3), (d, 4)}

f = {(a, 3), (b, 4), (c, 2), (d, 1)}

Weighted graph

is a graph G = (V ,E), along with a function w: E → R. The function w assigns a real number to every edge

Theorem: There is no smallest positive rational number. A proof by contradiction of the theorem starts by assuming which fact?

let r be the smallest rational number

Give the truth assignment that shows that the argument below is not valid: p V q ¬ q ------- p <-> q Note: Enter T or F in each blank

p: ["T"] q: [ "F"]

The propositional variables s and m represent the two propositions: s: It is sunny today. m: I will bring my umbrella. Give the logical expression that represents the statement: "Despite the fact that it is sunny today, I will bring my umbrella." Use symbols v, ^, and ~ as needed.

s^m

Select the statement that is equivalent to the statement: It is not true that x < 7

x ≥ 7

Theorem: For any real number x , x + | x − 5 | ≥ 5 In a proof by cases of the theorem, there are two cases. One of the cases is that x > 5. What is the other case? A) x<0 B) x≤5 C) none of these D) x≤0 E) x<5

x≤5

A={x∈Z: x is even} B={x∈Z: x is a prime number} D={5,7,8,12,13,15} Select the set corresponding to D-(A∪B). A) {5, 7, 13, 15} B) {13, 15} C) {8, 12, 15} D) {15}

{15}

A={x∈Z: x is a prime number} B={4,7,9,11,13,14} Select the set corresponding to A∩B. A) {7, 9, 11, 13} B) {4, 7, 9, 11, 13, 14} C) ϕ D) {7, 11, 13}

{7, 11, 13}

The domain for variable x is the set of all integers. Select the statement that is true.

∃ x ( x^2 < 1 )

The domain for x and y is the set of real numbers. Select the statement that is false.

∃ x ∀ y ( x + y ≥ 0 )

p = F, q = T, and r = T. Select the expression that evaluates to false.

∼ q