Exam 2

Question 1

Let p be “It is snowing.”
Let q be “I will study discrete math.”
“If it is snowing, then I will study discrete math.”
“It is snowing.”

Answer 1

Modus Ponens

Question 2

Let p be “it is snowing.”
Let q be “I will study discrete math.”
“If it is snowing, then I will study discrete math.”
“I will not study discrete math.”
“Therefore , it is not snowing.”

Answer 2

Modus Tollens

Question 3

Let p be “it snows.”
Let q be “I will study discrete math.”
Let r be “I will get an A.”
“If it snows, then I will study discrete math.”
“If I study discrete math, I will get an A.”
“Therefore , If it snows, I will get an A.”

Answer 3

Hypothetical Syllogism

Question 4

Let p be “I will study discrete math.”
Let q be “I will study English literature.”
“I will study discrete math or I will study English
literature.”
“I will not study discrete math.”
“Therefore , I will study English literature.

Answer 4

Disjunctive Syllogism

Question 5

Let p be “I will study discrete math.”
Let q be “I will visit Las Vegas.”
“I will study discrete math.”
“Therefore, I will study discrete math or I will visit Las Vegas.”

Answer 5

Addition

Question 6

Let p be “I will study discrete math.” Let q be
“I will study English literature.” “I will study
discrete math and English
literature”
“Therefore, I will study discrete math.”

Answer 6

Simplification

Question 7

Let p be “I will study discrete math.”
Let q be “I will study English literature.”
“I will study discrete math.”
“I will study English literature.”
“Therefore, I will study discrete math and I will
study English literature.”

Answer 7

Conjunction

Question 8

Let p be “I will study discrete math.”
Let r be “I will study English literature.”
Let q be “I will study databases.”
“I will not study discrete math or I will study English literature.”
“I will study discrete math or I will study databases.”
“Therefore, I will study databases or I will study English literature.”

Answer 8

Resolution

Question 9

Answer 9

Question 10

Answer 10

Question 11

Answer 11

Question 12

Translate the statement
∀x(C(x) ∨ ∃y(C(y) ∧ F(x, y))) into English, where C(x) is “x has a computer,” F(x, y) is “x and y are friends,” and the domain for both x and y consists of all students in your school

Answer 12

The statement says that for every student x in your
school, x has a computer, or there is a student y such that y has a computer and x and y are friends. In other words, every student in your school has a computer or has a friend who has a computer.

Question 13

Let Q(x, y, z) be the statement “x + y = z.” What are the
truth values of the statements
∀x∀y∃zQ(x, y, z) and ∃z∀x∀yQ(x, y, z), where the domain
of all variables consists of all real numbers?

Answer 13

“For every two integers, if these integers are both positive, then the sum of these integers is positive.”
“For all positive integers x and y, x + y is positive.”
Consequently, we can express this statement as
∀x∀y((x > 0) ∧ (y > 0) → (x +y > 0)),

Question 14

Use a proof by contradiction to give a proof that √2 is irrational.

Answer 14