Comp B14 Final

Use De Morgan's law to select the statement 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. Every student did not get an A on the test Every student got an A on the test. There is a student who got an A on the test.

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

Note that some arrows have heads at both ends. That means that both elements are related to each other. Note also that the self-loops do not have arrow heads. Enter H ο G as a sequence of ordered pairs [e.g. (2,3),(3,4),(4,5) ] Hint: There are 3 ordered pairs.

(1,4),(2,4),(2,2)

For f:{0,1}4→{0,1}4 f(x) is obtained by removing the second bit from x and placing the bit at the end of the string. For example, f(1011) = 1110. Select the correct value for f-1(0101).

0110

A directed graph G has 5 vertices, numbered 1 through 5. The 5 x 5 matrix A is the adjacency matrix for G. There is exactly one walk of length 4 from vertex 4 to vertex 2. What is the middle vertex in the walk? ⟨4,*,?,*,2⟩

1

A group of 15 friends are lining up in a row for a photograph. How many ways are there to line up the 15 friends if 5 want to be next to each other in the photo?

11! * 5!

f is a function that maps 12-bit binary strings to 11-bit binary strings. For x∈{0,1}12, f(x) is the string x with the last 1 bit(s) removed. Which property best describes the function f? Bijection 2-to-1 correspondence 4-to-1 correspondence 8-to-1 correspondence 16-to-1 correspondence 32-to-1 correspondence

2-to-1

The domain of relation R is the set of all non-negative integers. x is related to y if ⌊x/3⌋ = ⌊y/3⌋. The equivalence class R defines a partition over the set of all non-negative integers. How many elements are in each set in the partition? 0 1 2 3 4 Each set in the partition is infinite.

3

A group of 84 students were surveyed about the courses they were taking at their college with the following results: 52 students said they were taking Math. 48 students said they were taking English. 45 students said they were taking History. 30 students said they were taking Math and English. 34 students said they were taking Math and History. 25 students said they were taking English and History. 18 students said they were taking all three courses. How many students took none of the courses?

52+48+45-30-34-25+18=74 84-74=10

A person's birth date consists of the month, day, and year in which that person was born. The domain for a relation R is a group of people. No two people in the group have the same birth date. A person x is related to person y under the relation if x's birth date is earlier than y's birth date. Which description correctly characterizes the relation? A strict order but not a total order. A partial order and a total order. A partial order but not a total order. A strict order and a total order.

A strict order and a total order

The domain of a relation is the set of all positive integers. x is related to y if ⌊x/2⌋<⌊y/2⌋. Select the description that correctly characterizes the relation. Neither a partial order nor a strict order A partial order but not a total order A strict order but not a total order A partial order and a total order A strict order and a total order

A strict order but not a total order

Find the minimum spanning tree on the graph above using Prim's algorithm. Select all of the edges below that are included in the minimum cost spanning tree?

AC BE BC DE

Solve the recurrence relation (use n as the index variable) gn = -13g_n-1 + -42g_n-2 where g0 = -12 and g1=74

Answer: gn = -2(-7)^n - 10(-6)^n Steps: gn = -13g_n-1 - 42g_n-2 x^n = -13x^n-1 - 42x^n-2 x^2 = -13x - 42 x=-7 x=-6 gn= a0 (-7)^n +a1 (-6)^n -12 = a0 + a1 74 = -7a0 - 6a1 a0 = -2, a1 = -10 gn = (-2)(-7)^n -10(-6)^n

Use De Morgan's Laws to write a statement that is equivalent to the following statement: "It is not true that North Dakota and East Dakota are both states." - Either East Dakota is a state, or North Dakota is not a state. - Either North Dakota is not a state, or East Dakota is not a state. - Either North Dakota is a state, or East Dakota is not a state. - North Dakota is not a state, and East Dakota is not a state. - North Dakota is not a state, and East Dakota is a state.

Either North Dakota is not a state, or East Dakota is not a state.

Choose the set identity that is used to justify the following. (B - A) ∪ (B - A) = (B - A)

Idempotent Law

The domain of a relation is the set of all positive integers. x is related to y if ⌊x/2⌋ ≤ ⌊y/2⌋. Select the description that correctly characterizes the relation. Neither a partial order nor a strict order A partial order but not a total order A strict order but not a total order A partial order and a total order A strict order and a total order

Neither a partial order nor a strict order

The domain of a relation R is the set of real numbers. xRy if ⌈x⌉ ≤ ⌈y⌉. Select the description that accurately describes relation R. Anti-reflexive Symmetric Neither symmetric nor anti-symmetric Anti-symmetric

Neither symmetric nor anti-symmetric

A certain disease has an incidence rate of 0.5%. If the false negative rate is 7% and the false positive rate is 2%, compute the probability that a person who tests positive actually has the disease.

P(D) = 0.005 P(~D) = 0.995 P(~T|D) = 0.07 P(T|D) = 0.93 P(T|~D) = 0.02 P(~T|~D) = 0.98 (0.93 x 0.005)/((0.93 x 0.005+0.02*0.995)) 0.18940937

The domain of relation R is Z x Z. (a, b) is related to (c, d) if a-b = c-d. Which statement correctly characterizes the relation R? R is an equivalence relation. R is not an equivalence relation because R is not reflexive. R is not an equivalence relation because R is not symmetric. R is not an equivalence relation because R is not transitive.

R is an equivalence relation.

The domain of relation R is Z x Z. (a, b) is related to (c, d) if a≤c and b≤d. Which statement correctly characterizes the relation R? R is an equivalence relation. R is not an equivalence relation because R is not reflexive. R is not an equivalence relation because R is not symmetric. R is not an equivalence relation because R is not transitive.

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

The domain of relation R is the set of all integers. x is related to y if |x-y| ≤ 1. Which statement correctly characterizes the relation R? R is an equivalence relation. R is not an equivalence relation because R is not reflexive. R is not an equivalence relation because R is not symmetric. R is not an equivalence relation because R is not transitive

R is not an equivalence relation because R is not transitive.

Choose ALL the valid arguments. p ∨ q p ------------ ∴ q p ↔ q p ∨ q ------------ ∴ p p q ------------ ∴ p ↔ q p ∨ q ~q ------------ ∴ p ↔ q (p ∧ q) → r ------------ ∴ (p ∨ q) → r (p ∨ q) → r ------------ ∴ (p ∧ q) → r

Second, third, and last one.

The domain for variable x is the set of all integers. Select the correct rule to replace (?) in the proof segment below: 1. ∀x(P(x)∧Q(x)) Hypothesis 2. 3 is an integer Hypothesis 3. P(3)∧Q(3) (?) Existential instantiation Universal generalization Existential generalization Universal instantiation

Universal Instantiation

Find: 9∑i=02(5)i

Use formula

A fair coin is flipped 13 times. What is the probability that the first 3 of the 13 flips come up heads AND the last 5 flips come up tails? What is the probability that the first 3 of the 13 flips come up heads OR the last 5 flips come up tails? What is the probability that the first 3 of the 13 flips come up heads AND the last 11 flips come up tails? What is the probability that the first 3 of the 13 flips come up heads OR the last 11 flips come up tails?

What is the probability that the first 3 of the 13 flips come up heads AND the last 5 flips come up tails? (1/2)^3(1/2)^5 What is the probability that the first 3 of the 13 flips come up heads OR the last 5 flips come up tails? (1/2)^3+(1/2)^5-(1/2)^3(1/2)^5 What is the probability that the first 3 of the 13 flips come up heads AND the last 11 flips come up tails? 0 What is the probability that the first 3 of the 13 flips come up heads OR the last 11 flips come up tails? (1/2)^3+(1/2)^11

How many 8-character passwords can be made using the characters {a, b, c, d, e, f, g, h, i, j, k,l} (12 characters) if the first character must be one of {c,d,f,j,k} and ...a) Repetitions ARE allowed b) Repetitions are NOT allowed

a) Repetitions allowed: 5 * 12^7 b) Repetitions not allowed: 5 * 11(10)(9)(8)(7)(6)(5)

The general solution for a recurrence relation is bn=a_0 * 4^n+a_1*(−1)^n, and the initial values are b_0=4, b_1=1. Select the two linear equations that must be solved to find the correct values for constants a_0 and a1_. 4a_0 + 4a_1 = 4 4a0 - a1 = 1 a0 + a1 = 4 4a0 - 4a1 = 1 a0 + a1 = 4 4a0 - a1 = 1 4a0 + a1 = 4 4a0 - 4a1 = 1

a0 + a1 = 4 4a0 - a1 = 1

Assume the world's population increases by 6.5% every decade. Define gn to be the population after n decades. Enter the value in the statement below that represents the correct recurrence relation that describes the sequence {gn} . gn = ·gn-1

g_n = 1.065 * g_n-1

A basket holds a set of balls. Each ball is one of the following colors: {orange, indigo, red, yellow, blue}. What is the minimum number of balls that must be in the basket in order to guarantee that there are at least 19 balls of the same color?

k(b-1)+1 5(19-1)+1 = 91

A fair coin is flipped 7 times. What is the probability that exactly 3 of the 7 flips come up heads?

nCr(7,3)/2^7 = 35/128

How many strings of length 7 over the alphabet { a, b, c, d, e, f, g, h, i, j, k, l } (12 characters) have exactly 4 a's?

nCr(7,4) * 11^3 = 46585

What is the coefficient of x^4*y^4 in (3x - 2y)^8?

nCr(8,4) * 3^4 * (-2)^4 = 90720

Select the truth assignment that shows that the argument below is not valid: p ∨ q ~q ------ ∴ p ↔ q p = T, q = F p = F, q = F p = F, q = T p = T, q = T

p = T, q = F

Given the laws shown above, show that, for the conditional proposition p → q, its converse is equivalent to its inverse. Start with the converse in the upper row of the left column and show each step and its justification.

q → p Converse of Conditional ¬q v p Conditional identities p v ¬q Commutative Laws ¬¬p v ¬q Double negation law ¬p → ¬q Conditional identities

The domain of discourse for x and y is the set of students in a course. Define the predicate: A(x): student x got an 'A' Select the logical expression that is equivalent to: "Exactly one student got an 'A' in the course." ∃x ∀y (A(x) ∧ ((x ≠ y) → ¬A(y))) ∃x A(x) ∃x ∀y (A(x)∧ ¬A(y)) ∃x ∀y (A(x)→((y≠x)∧ ¬A(y)))

x ∀y (A(x) ∧ ((x ≠ y) → ¬A(y)))

Which relation on the set {1, 2, 3, 4} is a partial order? { (1, 2), (2, 3), (1, 3), (4, 3) } { (1, 2), (2, 3), (1, 3), (3, 4), (1, 1), (2, 2), (3, 3), (4, 4) } { (1, 2), (2, 3), (1, 3), (4, 3), (1, 1), (2, 2), (3, 3), (4, 4) } { (1, 2), (2, 3), (1, 3), (3, 4) }

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

For the set A = {1, 2, 3, 4}. Check all the true statements about the power set, P(A). ∅⊆P(A) {4}⊆P(A) |P(A)| = 4 {2,3}⊆P(A) ∅∈P(A) {2,3}∈P(A)

{2,3}∈P(A) ∅⊆P(A) ∅∈P(A)

Given the following sets A={t,u} B={b,n,q,r,s,u,w} C={b,d,g,h,p,r,z} Express the following set in roster notation. Express elements of Cartesian products as strings - for example, if the Cartesian product were {(a,b), (a,c)} you would express it as {ab, ac}. (Note the outside curly braces are already provided.) A×(B∩C) =

{tb, tr, ub, ur}

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)) ∃x (¬S(x)↔A(x)) ∃x (¬S(x)∧A(x)) ∃x (A(x)→¬S(x))

∃x (¬S(x)∧A(x))

Select the logical expression that is equivalent to: ¬∀x ∃y (P(x)∧Q(x,y)) ∀y ∃x (¬P(x)∨¬Q(x,y)) ∃y ∀x (¬P(x)∨¬Q(x,y)) ∃x ∀y (¬P(x)∨¬Q(x,y)) ∀x ∃y (¬P(x)∨¬Q(x,y))

∃x ∀y (¬P(x)∨¬Q(x,y))

The domain for variables x and y is the set {1, 2, 3}. The table below gives the values of P(x, y) for every pair of elements from the domain. For example, P(2, 3) = F because the value in row 2, column 3, is F. Note: The first variable in P(x, y) is the row number and the second is the column number. Select the statement that is FALSE. ∃y ∀x P(x,y) ∃x ∀y P(x,y) ∀x ∃y P(x,y) ∀y ∃x P(x,y)

∃y ∀x P(x,y)

How many 12-bit strings begin with "01000" or "000"?

2^7 + 2^9

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: "Everyone who studied for the test received an A on the test" ∀x (S(x)∧A(x)) ∀x (S(x)↔A(x)) ∀x (S(x)→A(x)) ∀x (A(x)→S(x))

∀x (S(x)→A(x))

The domain for variable x is the set 1, 2, 3. The table below gives the values of P(x,y) for every pair of elements from the domain. For example, P(2, 3) = T because the value in row 2, column 3, is T. Note: The first variable in P(x, y) is the row number and the second is the column number. Define the predicate: Select the statement that is TRUE. ∀x ∀y ((x ≠ y) → P(x,y)) ∀x ∀y ((x ≠ y) ∨ P(x,y)) ∀x ∀y P(x,y) ∀x ∀y ((x ≠ y) ∧ P(x,y))

∀x ∀y ((x ≠ y) → P(x,y))

Complete the mathematical expression that is equivalent to the sum: ∑(10+6j) from j=0 to k+1

∑j=0(10+6j) from j=0 to k + 10 + 6(k+1)

Given the recurrence recurrence relation: g_n = 9g_n-1 + 10g_n-2 where g_0 = -18 g_1 = -70 What is g_2?

-810

You are stringing 19 beads on a string. The beads come in 6 colors. Without having any two adjacent beads be the same color, how many different ways can you string the beads?

6 * 5^18

Graph G is defined by the arrow diagram below. Select the statement about G that is false. G has a circuit of length 4 G has a cycle of length 3 G has a cycle of length 4 G has a circuit of length 3

G has a cycle of length 4

Find: 50∑i=0(11+8i)

Use formula