Discrete Math - Chapter 1

What is the difference between 'OR' and 'XOR'?

'XOR', or exclusive OR would yield false for the case where the propositions in question both yield T, whereas with 'OR' it would yield true

The predicate T is defined as: T(x, y, z): (x + y)^2 = z Select the proposition that is true. a. T(4, 1, 5) b. T(4, 1, 25) c. T(1, 1, 1) d. T(4, 0 2)

b. T(4, 1, 25)

Select the statement that is equivalent to the statement: It is not true that x < 7 a. x > 7 b. x ≥ 7 c. x ≤ 7 d. x = 7

b. x ≥ 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: "Everyone who studied for the test received an A on the test." a. ∀x(A(x) → S(x)) b. ∀x(S(x) → A(x)) c. ∀x(S(x) ∧ A(x)) d. ∀x(S(x) ↔ A(x))

b. ∀x(S(x) → A(x))

The domain for variable x is the set of all integers. Select the statement that is true. a. ∃x(3x = 1) b. ∃x(x^2 < 1) c. ∀x(x^2 = 1) d. ∃x(x^2 < 0)

b. ∃x(x^2 < 1) when x = 0

The domain for x and y is the set of real numbers. Select the statement that is false. a. ∀x∃y(x + y ≥ 0) b. ∃x∀y(x + y ≥ 0) c. ∀x∃y(xy ≥ 0) d. ∃x∀y(xy ≥ 0)

b. ∃x∀y(x + y ≥ 0)

What is the word for "if and only if"

biconditional

Select the correct rule to replace (?) in the proof segment below: 1. ¬p ∨ q Hypothesis 2. ¬¬p Hypothesis 3. q (?) a. Simplification b. Hypothetical syllogism c. Disjunctive syllogism d. Resolution

c. Disjunctive syllogism

Select the true statement. a. For any real number x, x > 5 implies that x ≥ 6. b. For any real number x, x ≥ 5 implies that x ≥ 6. c. For any real number x, x > 5 implies that x ≥ 5. d. For any real number x, x ≥ 5 implies that x > 5.

c. For any real number x, x > 5 implies that x ≥ 5.

41) Select the truth assignment that shows that the argument below is not valid: p ∨ q ¬q ------- p ↔ q a. p = T q = T b. p = F q = T c. p = T q = F d. p = F q = F

c. p = T q = F

27) 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." a. ∃x(A(x) → ¬S(x)) b. ∃x(¬S(x) → A(x)) c. ∃x(¬S(x) ∧ A(x)) d. ∃x(¬S(x) ↔ A(x))

c. ∃x(¬S(x) ∧ A(x))

Which statement is false? a. 4 | − 16 b. 2 ∤ 5 c. 7 | 0 d. 1 ∤ 5

d. 1 ∤ 5

Given the conditional statement, p -> q, what is the form of the inverse?

-p -> -q

Given the conditional statement, p -> q, what is the form of the contrapositive?

-q -> -p

What is the definition of a proposition?

A declarative sentence that is true or false, but not both

What is another word for the logical connective "and"?

Conjunction

What is the term for a proposition that is always false?

Contradiction

What is another word for the logical connective "or"?

Disjunction

What is the type of quantification represented by the phrase, "there exists an x such that"

Existential quantification

What is the term for an incorrect argument?

Fallacy

With nested quantifiers, does the order of the terms matter?

If they are of the same type (both existential or both universal) it doesn't matter. If they are of different types, it does matter.

What is another word for 'conditional statement'?

Implication

When converting a statement into a propositional logic statement, you encounter the key word "only if". Should you flip the order of the statement or not?

No, don't flip the order around

What is a predicate?

Statement involving variables where the truth value is not known until a variable value is assigned

What is the term for a proposition that is always true?

Tautology

How do you determine if two statements are logically equivalent?

The p <-> q biconditional is a tautology

What is the type of quantification represented by the phrase, "there exists only one x such that"

Uniqueness quantifier (represented with !)

What is the type of quantification represented by the phrase, "for every x"

Universal quantification

When is a conditional statement false?

When the hypothesis is True, but the conclusion is False

When converting a statement into a propositional logic statement, you encounter the key word "if". Should you flip the order of the statement or not?

Yes, flip the statement around

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) (?) a. Universal instantiation b. Universal generalization c. Existential instantiation d. Existential generalization

a. Universal instantiation

Select the correct expression for (?) in the proof segment below: 1. p → r Hypothesis 2. p ∧ q Hypothesis 3. (?) Simplification, 2 4. r Modus Tollens, 1, 3 a. p b. q c. p ∨ q d. p ∧ q

a. p

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 true. a. ∀x(Q(x) → P(x)) b. ∀x(P(x) → Q(x)) c. ∀x(P(x) ∧ Q(x)) d. ∀x(P(x) ∨ Q(x))

a. ∀x(Q(x) → P(x))

Select the logical expression that is equivalent to: ¬∀x(¬P(x) ∨ Q(x)) a. ∃x(P(x) ∧ ¬Q(x)) b. ∃x(¬P(x) ∨ Q(x)) c. ∀x(P(x) ∨ ¬Q(x)) d. ∀x(¬P(x) ∧ Q(x))

a. ∃x(P(x) ∧ ¬Q(x))

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

a. ∃x∀y(¬P(x) ∨ ¬Q(x, y))

Use De Morgan's law to select the statement that is logically equivalent to: "It is not true that there was a student who was absent yesterday." a. Every student was absent yesterday. b. Every student was not absent yesterday. c. Some student was absent yesterday. d. At least one student was not absent yesterday.

b. Every student was not absent yesterday.

Select the correct rule to replace (?) in the proof segment below: 1. ¬p → ¬q Hypothesis 2. ¬q → r Hypothesis 3. ¬p → r (?) a. Modus ponens b. Hypothetical syllogism c. Disjunctive syllogism d. Conditional identity

b. Hypothetical syllogism

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

b. Universal generalization

-7 is an odd number because -7 = 2k+1 for some integer k. 34 is an even number because 34 = 2j for some integer j. Select the correct values for k and j. a. k = -3, j = 17 b. k = -4 j = 17 c. k = -3, j = -17 d. k = -4 j = -17

b. k = -4 j = 17

Select the truth assignment that shows that the argument below is not valid: p → q q ----- p a. p = T q = T b. p = F q = T c. p = T q = F d. p = F q = F

b. p = F q = T

One way to show that the number -0.33 is rational is to show that -0.33 = x/y, where x and y are integers and y is non-zero. Select a pair of values for x and y to show that -0.33 is rational. a. x = 33, y = 100 b. x = 33, y = -100 c. x = 100, y = 33 d. x = 100, y = -33

b. x = 33, y = -100

The domain for variable x is the set of all integers. Select the statement that is false. a. ∀x(x^2 ≠ 5) b. ∀x(x^2 ≥ x) c. ∀x(x^2 > x) d. ∃x( sqrt(x) = x)

c. ∀x(x^2 > x)

Select the logical expression that is equivalent to: ¬∃x(P(x) ∧ Q(x)) a. ∃x(¬P(x) ∨ ¬Q(x)) b. ∃x(¬P(x) ∧ ¬Q(x)) c. ∀x(¬P(x) ∨ ¬Q(x)) d. ∀x(¬P(x) ∧ ¬Q(x))

c. ∀x(¬P(x) ∨ ¬Q(x))

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))

c. ∃x(¬P(x) ∧ Q(x))

The domain of discourse for x and y is the set of employees at a company. Miguel is one of the employees at the company. Define the predicate: N(x, y): x earns more than y Select the logical expression that is equivalent to: "Exactly one person earns more than Miguel." a. ∃𝑥 N(x,Miguel) b. ∃x∀y(N(x,Miguel) ∧ ¬N(y,Miguel)) c. ∃x∀y(N(x,Miguel) ∧ ((y ≠ x) → ¬N(y,Miguel))) d. ∃x∀y(N(x,Miguel) → ((y ≠ x) → ¬N(y,Miguel)))

c. ∃x∀y(N(x,Miguel) ∧ ((y ≠ x) → ¬N(y,Miguel)))

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. P 1 2 3 1 T T T 2 T F F 3 F T F Select the statement that is false. a. ∃x∀y P(x, y) b. ∀x∃y P(x, y) c. ∃y∀x P(x, y) d. ∀y∃x P(x, y)

c. ∃y∀x P(x, y)

Select the true statement. a. 2 is composite b. -2 is composite c. -5 is prime d. 5 is prime

d. 5 is prime

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

d. Existential generalization

Which rule is used in the argument below? Alice is a student in the class. Alice got an A on the test and did not study. Therefore, there is a student in the class who got an A on the test and did not study. a. Universal instantiation b. Universal generalization c. Existential instantiation d. Existential generalization

d. Existential generalization

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." a. Every student did not get an A on the test. b. There is a student who got an A on the test. c. Every student got an A on the test. d. There is a student who did not get an A on the test.

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

Select the statement that is false. a. x = 2 implies x ≤ 2. b. x < 2 implies that x ≤ 2. c. x = 2 implies that x ≥ 2. d. x < 2 implies that x ≥ 2.

d. x < 2 implies that x ≥ 2.

Select the correct expression for (?) in the proof segment below: 1. (p ∨ q) → r Hypothesis 2. ¬r Hypothesis 3. (?) Modus Tollens, 1, 2 a. ¬p b. ¬q c. ¬p ∨ ¬q d. ¬(p ∨ q)

d. ¬(p ∨ q)

The domain of discourse for x and y is the set of employees at a company. Define the predicate: V(x): x is a manager N(x, y): x earns more than y Select the logical expression that is equivalent to: "Every manager earns more than every employee who is not a manager." a. ∀x∀y (M(x, y) → (V(x) → ¬V(y))) b. ∀x∀y (V(x) ∧ ¬V(y)¬V(y) ∧ M(x, y)) c. ∀x∀y ((V(x) ∧ ¬V(y)) → M(x, y)) d. ∀x∀y M(V(x), ¬V(y))

d. ∀x∀y M(V(x), ¬V(y))

36) 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. P 1 2 3 1 T T T 2 T F F 3 F T F Select the statement that is false. a. ∃x∃yP(x, y) b. ∃yP(2, y) c. ∀yP(1, y) d. ∃y¬P(1, y)

d. ∃y¬P(1, y)

What is the form of an implication?

hypothesis/premise -> conclusion/consequence

Given the conditional statement, p -> q, what is the form of the converse?

q -> p