Question 1

Enter all propositions from the following statements in the blank space.

A: "The moon is made of green cheese."

B: "2n10"

C: "You get an A on CS250 final exam."

D: "5+7=10"

E: C^D

F: (BC) A

G: (C A) D

Question 2

Enter all compound propositions from the statements in the blank space.

A: "The moon is made of green cheese."

B: "2n10"

C: "You get an A on CS250 final exam."

D: "5+7=10"

E: C^D

F: (BC) A

G: (C A) D

Question 3

Let p and q be the propositions The election is decided and The votes have been counted, respectively. Express each of these compound propositions as an English sentence.

p q

q p

p q

Group of answer choices

1. a. The election is not decided, and the votes have been counted.

b. If the votes have been counted, then the election is decided.

c. The election is decided if and only if the votes have been counted.

2. a. The election is decided, and the votes have been counted.

b. If the votes have been counted, then the election is decided.

c. The election is decided if and only if the votes have been counted.

3. a. The election is not decided, and the votes have been counted.

b. If the votes have been counted, then the election is not decided.

c. The election is decided if and only if the votes have been counted.

4. a. The election is not decided, and the votes have been counted.

b. If the votes have been counted, then the election is decided.

c. The election is not decided if and only if the votes have been counted.

Question 4

Let p, q, and r be the following propositions.

p : You get an A on the final exam.

q : You do every exercise in this book.

r : You get an A in this class.

Express the following English sentence as a compound proposition.

You get an A in this class, but you do not do every exercise in this book.

Group of answer choices

1. a. rq

2. b. r(q)

3. c. rq

Question 5

Most compound propositions are constructed using paratheses to specify the order in which logical operators are to be applied. However, the convention of the order is shown in the following table when no parentheses are used.

Operator
Precedence	1	2	3	4	5

Determine true or false, suppose p, q, and r are propositions.

A. p q r means p (q r).

B. p q r means (p q) r.

Group of answer choices

1. A. True

B. False

2. A. False

B. False

3. A. True

B. True

4. A. False

B. True

Question 6

Determine whether each of these conditional/biconditional statements is true or false.

If 1 + 1 = 3, then unicorns exist.

1 + 1 = 2 if and only if 2 + 3 = 4.

Group of answer choices

1. true.

true.

2. true.

false.

3. false.

true.

4. false.

false.

Question 7

Consider that the original statement as " If it snows, then I stay at home ".

Identify the correct type of statement for following statements.

If I stay home, then it snows.

If it does not snow, then I stay home.

if I do not stay at home, then it does not snow.

Group of answer choices

1. inverse

converse

contrapositive

2. converse

contrapositive

inverse

3. converse

contrapositive

inverse

4. converse

not inverse

contrapositive

Question 8

Show that (p q) and q p are equivalent

(a) using a truth table. (b) using logical equivalences

Question 9

Prove the following statement.

The sum of an even integer and an odd integer is always odd

Question 10

Evaluate each of these expressions.

1 1000 (0 1011 1 1011)

(0 1111 1 0101) 0 1000

(0 1010 1 1011) 0 1000

(1 1011 0 1010) (1 0001 1 1011)

Group of answer choices

a. 11001

b. 01101

c. 11001

d. 11011

a. 11000

b. 01101

c. 11001

d. 11011

a. 11000

b. 01111

c. 11001

d. 11011

a. 11100

b. 01101

c. 11001

d. 11011

Question 11

Use De Morgans laws to find the negation of each of the following statements.

Kwame will take a job in industry or go to graduate school.

Rita will move to Oregon and Washington

Question 12

Use symbolic manuputation method to show that (p q) (p r) and p (q r) are logically equivalent

Question 13

Let N (x) be the statement x has visited North Dakota, where the domain consists of the students in your school. Express each of these quantifications in English.

xN (x)

xN (x)

xN (x)

Question 14

Translate these statements into English, where R(x) is x is a rabbit and H (x) is x hops and the domain consists of all animals.

x(R(x) H (x))

x(R(x) H (x))

Question 15

Determine the truth value of each of these statements if the domain consists of all real numbers.

x(x³ = 1)

x(2x > x)

x((x)² = x²)

Group of answer choices

1.a. true

b. false

c. true

2. a. true

b. true

c. true

3. a. true

b. false

c. false

4. a. false

b. false

c. true

Question 16

Let P (x, y) be the statement Student x has taken class y, where the domain for x consists of all students in your class and for y consists of all computer science courses. Translate each of the following using English.

xy P (x, y)

yx P (x, y)

yx P (x, y)

Question 17

Let Q(x, y) be the statement student x has been a contestant on quiz show y. Express each of these sentences in terms of Q(x, y), quantifiers, and logical connectives, where the domain for x consists of all students at your school and for y consists of all quiz shows on television

There is a student at your school who has been a contestant on a television quiz show.

No student at your school has ever been a contestant on a television quiz show

Group of answer choices

1. xyQ(x, y)

xyQ(x, y)

2. xyQ(x, y)

xyQ(x, y)

3. xyQ(x, y)

xyQ(x, y)

4. xyQ(x, y)

xyQ(x, y)

Question 18

Prove that if n is an integer and 3n + 2 is even, then n is even using

a proof by contraposition.

a proof by contradiction.

Question 19

Prove that sqrt(2) is an irrational number.

p