DDS Test 1

P. The sky is not purple.

Determine whether each of the following sentences is a proposition. If the sentence is a proposition, then write its negation (NP = Not a Proposition, P = Proposition). The sky is purple.

NP.

Determine whether each of the following sentences is a proposition. If the sentence is a proposition, then write its negation. (NP = Not a Proposition, P = Proposition) Have a nice day.

a) False c) True e) False

Determine whether the following propositions are true or false: (a) 5 is an odd number and 3 is a negative number. (c) 8 is an odd number or 4 is not an odd number. (e) It is not true that 7 is an odd number or 8 is an even number.

t ∧ n

Express each English statement using logical operations ∨, ∧, ¬ and the propositional variables t, n, and m defined below. The use of the word "or" means inclusive or. t: The patient took the medication. n: The patient had nausea. m: The patient had migraines. Despite the fact that the patient took the medication, the patient had nausea.

n ∧ m

Express each English statement using logical operations ∨, ∧, ¬ and the propositional variables t, n, and m defined below. The use of the word "or" means inclusive or. t: The patient took the medication. n: The patient had nausea. m: The patient had migraines. The patient had nausea and migraines.

n ∨ m

Express each English statement using logical operations ∨, ∧, ¬ and the propositional variables t, n, and m defined below. The use of the word "or" means inclusive or. t: The patient took the medication. n: The patient had nausea. m: The patient had migraines. The patient had nausea or migraines.

I am going to a movie and not going to the party tonight.

Express the following compound propositions in English using the following definitions: p: I am going to a movie tonight. q: I am going to the party tonight. c) p ∧ ¬q

I am not going to the a movie or the party tonight.

Express the following compound propositions in English using the following definitions: p: I am going to a movie tonight. q: I am going to the party tonight. e) ¬(p ∧ q)

I am not going to a movie tonight.

Express the following compound propositions in English using the following definitions: p: I am going to a movie tonight. q: I am going to the party tonight. (a) ¬p

a) ¬(p → q) b) p ↔ ¬q

For each table, give a logical expression whose truth table is the same as the one given. (a) p q ? T T F T F T F T F F F F (b) p q ? T T F T F T F T T F F F

¬p ∧ ¬q ∧ r

Give a logical expression with variables p, q, and r that is true if p and q are false and r is true and is otherwise false (Use ∨, ∧, ¬).

Inverse: If she did not finish her homework, the she did not went to the party. Converse: If she went to the party, then she finished her homework. Contrapositive: If she did not went to the party, then she did not finish her homework.

Give the inverse, converse and contrapositive for each of the following statements: (a) If she finished her homework, then she went to the party.

False.

(T/F) The truth value of the logic expression ∀x∃y∃z (x^2 = y^2 + z^2) is true if the domain is the set of all positive integers.

a) ∀x ¬P(x) c) ∃x (¬P(x) ∨ ¬Q(x))

Apply De Morgan's law to each expression to obtain an equivalent expression in which each negation sign applies directly to a predicate. (i.e., ∃x (¬P(x) ∨ ¬Q(x)) is an acceptable final answer, but not ¬∃x P(x) or ∃x ¬(P(x) ∧ Q(x))). (Use ∨, ∧, ¬, →, ∃, ∀) (a) ¬∃x P(x) (c) ¬∀x (P(x) ∧ Q(x))

True and true.

Assume the propositions p, q, r, and s have the following truth values: p is false q is true r is false s is true What are the truth values for the following compound propositions? (a) ¬p (c) q ∧ s

a) B ∨ D ∨ M b) (B ∧ M) ∨ (B ∧ D) ∨ (M ∧ D) c) B ∨ (D ∧ M)

Consider the following pieces of identification a person might have in order to apply for a credit card (Use ∨, ∧, ¬): B: Applicant presents a birth certificate. D: Applicant presents a driver's license. M: Applicant presents a marriage license. Write a logical expression for the requirements under the following conditions: (a) The applicant must present either a birth certificate, a driver's license or a marriage license. (b) The applicant must present at least two of the following forms of identification: birth certificate, driver's license, marriage license. (c) Applicant must present either a birth certificate or both a driver's license and a marriage license.

B

Define the domain of discourse for variables x and y to be the set of runners in a race. Define the predicate B(x, y) to mean that x beats y in the race. Which of the following logical expressions is equivalent to the English statement that someone was beaten by everyone else in the race? a) ∀x∃y (y≠x → B(y,x)) b) ∃x∀y (y≠x → B(y,x))

a) ¬j → c c) ¬j ∧ ¬c e) ¬c

Define the following propositions (Use ∨, ∧, ¬, →): c: I will return to college. j: I will get a job. Translate the following English sentences into logical expressions using the definitions above: (a) Not getting a job is a sufficient condition for me to return to college. (c) I am not getting a job, but I am still not returning to college. (e) There's no way I am returning to college.

p → (s ∨ y)

Define the following propositions (Use ∨, ∧, ¬, →): s: a person is a senior y: a person is at least 17 years of age p: a person is allowed to park in the school parking lot (e) Being able to park in the school parking lot implies that the person is either a senior or at least 17 years old.

a) false c) true e) true g) false i) true

Determine the truth value of each expression below. The domain is the set of all real numbers. (a) ∀x∃y (xy > 0) (c) ∀x∀y∃z (z = (x - y)/3) (e) ∀x∀y (xy = yx) (g) ∀x∃y y^2 = x (i) ∃x ∃y (x^2 = y^2 ∧ x ≠ y)

P. It is not true that every prime number is even.

Determine whether each of the following sentences is a proposition. If the sentence is a proposition, then write its negation (NP = Not a Proposition, P = Proposition). Every prime number is even.

P. It's not a beautiful day.

Determine whether each of the following sentences is a proposition. If the sentence is a proposition, then write its negation (NP = Not a Proposition, P = Proposition). It's a beautiful day.

P. The patient does not have diabetes.

Determine whether each of the following sentences is a proposition. If the sentence is a proposition, then write its negation (NP = Not a Proposition, P = Proposition). The patient has diabetes.

Proposition. There is a patient who had migraines and was given the medication. False.

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates: P(x): x was given the placebo D(x): x was given the medication A(x): x had fainting spells M(x): x had migraines Suppose that there are five patients who participated in the study. The table below shows the names of the patients and the truth value for each patient and each predicate: P(x)D(x)A(x) M(x) Frodo T F F T Gandalf F T F F Gimli F T T F Aragorn T F T T Bilbo T T F F For each of the following quantified statements, indicate whether the statement is a proposition. If the statement is a proposition, give its truth value and translate the expression into English. (a) ∃x (M(x) ∧ D(x))

Proposition. Every patient who had migraines had fainting spells and vice versa. False.

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates: P(x): x was given the placebo D(x): x was given the medication A(x): x had fainting spells M(x): x had migraines Suppose that there are five patients who participated in the study. The table below shows the names of the patients and the truth value for each patient and each predicate: P(x)D(x)A(x) M(x) Frodo T F F T Gandalf F T F F Gimli F T T F Aragorn T F T T Bilbo T T F F For each of the following quantified statements, indicate whether the statement is a proposition. If the statement is a proposition, give its truth value and translate the expression into English. (e) ∀x (M(x) ↔ A(x))

Proposition. There is a patient who was given the medication that had neither fainting spells nor migraines True.

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates: P(x): x was given the placebo D(x): x was given the medication A(x): x had fainting spells M(x): x had migraines Suppose that there are five patients who participated in the study. The table below shows the names of the patients and the truth value for each patient and each predicate: P(x)D(x)A(x) M(x) Frodo T F F T Gandalf F T F F Gimli F T T F Aragorn T F T T Bilbo T T F F For each of the following quantified statements, indicate whether the statement is a proposition. If the statement is a proposition, give its truth value and translate the expression into English. (g) ∃x (D(x) ∧ ¬A(x) ∧ ¬M(x))

Not a proposition.

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates: P(x): x was given the placebo D(x): x was given the medication A(x): x had fainting spells M(x): x had migraines Suppose that there are five patients who participated in the study. The table below shows the names of the patients and the truth value for each patient and each predicate: P(x) D(x) A(x) M(x) Frodo T F F T Gandalf F T F F Gimli F T T F Aragorn T F T T Bilbo T T F F For each of the following quantified statements, indicate whether the statement is a proposition. If the statement is a proposition, give its truth value and translate the expression into English. (c) ∃x M(x) ∧ D(x)

∀x D(x). Negation: ¬∀x D(x). Applying De Morgan's Law: ∃x ¬D(x). English There was a patient that was not given the medication.

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates: P(x): x was given the placebo D(x): x was given the medication M(x): x had migraines Translate each statement into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until each negation operation applies directly to a predicate and then translate the logical expression back into English (Use ∨, ∧, ¬, →, ∃, ∀). Sample question: Some patient was given the placebo and the medication. ∃x (P(x) ∧ D(x)) Negation: ¬∃x (P(x) ∧ D(x)) Applying De Morgan's law: ∀x (¬P(x) ∨ ¬D(x)) English: Every patient was either not given the placebo or not given the medication (or both). (a) Every patient was given the medication.

∀x (D(x) ∨ P(x)). Negation: ¬∀x (D(x) ∨ P(x)). De Morgan's Law: ∃x (¬D(x) ∧ ¬P(x)). English: There was a patient that was neither given the medication nor the placebo.

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates: P(x): x was given the placebo D(x): x was given the medication M(x): x had migraines Translate each statement into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until each negation operation applies directly to a predicate and then translate the logical expression back into English. Sample question: Some patient was given the placebo and the medication. (Use ∨, ∧, ¬, →, ∃, ∀). ∃x (P(x) ∧ D(x)) Negation: ¬∃x (P(x) ∧ D(x)) Applying De Morgan's law: ∀x (¬P(x) ∨ ¬D(x)) English: Every patient was either not given the placebo or not given the medication (or both). (b) Every patient was given the medication or the placebo or both.

∃x (D(x) ∧ M(x)). Negation: ¬∃x (D(x) ∧ M(x)). De Morgan's Laws: ∀x (¬D(x) ∨ ¬M(x)). English: Every patient was not given the medication or did not have migraines or both.

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates: P(x): x was given the placebo D(x): x was given the medication M(x): x had migraines Translate each statement into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until each negation operation applies directly to a predicate and then translate the logical expression back into English. Sample question: Some patient was given the placebo and the medication. (Use ∨, ∧, ¬, →, ∃, ∀). ∃x (P(x) ∧ D(x)) Negation: ¬∃x (P(x) ∧ D(x)) Applying De Morgan's law: ∀x (¬P(x) ∨ ¬D(x)) English: Every patient was either not given the placebo or not given the medication (or both). (c) There is a patient who took the medication and had migraines.

∀x ∀y F(x, y). Negation: ¬∀x ∀y F(x, y). De Morgan's Law: ∃x ∃y ¬F(x, y). English: Someone is an enemy of someone.

The domain for variables x and y is a group of people. The predicate F(x, y) is true if and only if x is a friend of y. For the purposes of this problem, assume that for any person x and person y, either x is a friend of y or x is an enemy of y. Therefore, ¬F(x, y) means that x is an enemy of y. Translate each statement into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until the negation operation applies directly to the predicate and then translate the logical expression back into English. (Use ∨, ∧, ¬, →, ∃, ∀). (a) Everyone is a friend of everyone.

∀x (P(x) → M(x)). Negation: ¬∀x (P(x) → M(x)). De Morgan's Law: ∃x (P(x) ∧ ¬M(x)). English: There is a patient that was given the placebo and did not have migraines.

In the following question, the domain of discourse is a set of male patients in a clinical study. Define the following predicates: P(x): x was given the placebo D(x): x was given the medication M(x): x had migraines Translate each statement into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until each negation operation applies directly to a predicate and then translate the logical expression back into English. Sample question: Some patient was given the placebo and the medication. (Use ∨, ∧, ¬, →, ∃, ∀). ∃x (P(x) ∧ D(x)) Negation: ¬∃x (P(x) ∧ D(x)) Applying De Morgan's law: ∀x (¬P(x) ∨ ¬D(x)) English: Every patient was either not given the placebo or not given the medication (or both). (d) Every patient who took the placebo had migraines. (Hint: you will need to apply the conditional identity, p → q ≡ ¬p ∨ q.)

a) ∃x (T(x) ∧ E(x)) b) ∀x (E(x) → T(x)) c) ∀x (T(x) → E(x)) d) ∃x (E(x) ∧ ¬T(x))

In the following question, the domain of discourse is a set of students at a university. Define the following predicates (Use ∨, ∧, ¬, →, ∃, ∀): E(x): x is enrolled in the class T(x): x took the test Translate the following English statements into a logical expression with the same meaning. (a) Someone took the test who is enrolled in the class. (b) All students enrolled in the class took the test. (c) Everyone who took the test is enrolled in the class. (d) At least one student who is enrolled in the class did not take the test.

a) ∃x ¬B(x) c) (¬B(Sam) ∧ T(Sam)) e) ∀x (T(x) → B(x))

In the following question, the domain of discourse is the set of employees at a company. One of the employees is named Sam. Define the following predicates (Use ∨, ∧, ¬, →, ∃, ∀): T(x): x is a member of the executive team B(x): x received a large bonus Translate the following English statements into a logical expression with the same meaning. (a) Someone did not get a large bonus. (c) Sam did not get a large bonus even though he is a member of the executive team. (e) Every executive team member got a large bonus.

Every employee who is on the board of directors earns more than $100,000.

In the following question, the domain of discourse is the set of employees of a company. Define the following predicates: A(x): x is on the board of directors E(x): x earns more than $100,000 W(x): x works more than 60 hours per week Translate the following logical expressions into English: (a) ∀x (A(x) → E(x))

Every employee who works more than 60 hours per week earns more than $100,000.

In the following question, the domain of discourse is the set of employees of a company. Define the following predicates: A(x): x is on the board of directors E(x): x earns more than $100,000 W(x): x works more than 60 hours per week Translate the following logical expressions into English: (c) ∀x (W(x) → E(x))

Every employee who earns more than $100,000 is on the board of directors or works more than 60 hours per week.

In the following question, the domain of discourse is the set of employees of a company. Define the following predicates: A(x): x is on the board of directors E(x): x earns more than $100,000 W(x): x works more than 60 hours per week Translate the following logical expressions into English: (e) ∀x (E(x) → (A(x) ∨ W(x)))

a) false c) true e) false (x = 0)

In this problem, the domain of discourse is the set of all integers. Which statements are true? If an existential statement is true, give an example. If a universal statement is false, give a counterexample. (a) ∃x (x + x = 1) (c) ∀x (x^2 − x ≠ 1) (e) ∀x (x^2 > 0)

a) Inclusive: True. Exclusive: True. c) Inclusive: True. Exclusive: False. e) Inclusive: True. Exclusive: False.

Indicate whether each statement is true or false, assuming that the "or" in the sentence means the inclusive or. Then indicate whether the statement is true or false if the "or" means the exclusive or. (a) February has 31 days or the number 5 is an integer. (c) 20 nickels are worth one dollar or whales are mammals. (e) January has exactly 31 days or April has exactly 30 days.

a) Proposition, true c) Proposition, true e) Proposition, false

Predicates P and Q are defined below. The domain of discourse is the set of all positive integers. P(x): x is prime Q(x): x is a perfect square (i.e., x = y^2, for some integer y) Indicate whether each logical expression is a proposition. If the expression is a proposition, then give its truth value. (a) ∃x Q(x) (c) ∀x Q(x) ∨ P(3) (e) ∀x (¬Q(x) ∨ P(x))

a) proposition, false; c) proposition, true; e) proposition, true; g) proposition, true

Predicates P, T, and E are defined below. The domain of discourse is the set of all positive integers. P(x): x is even T(x, y): 2^x = y E(x, y, z): x^y = z Indicate whether each logical expression is a proposition. If the expression is a proposition, then give its truth value. (a) P(3) (c) T(5, 32) (e) E(6, 2, 36) (g) P(3) ∨ T(5, 32)

The expression is true if at least one propositional variable is true. It is false when all of the propositional variables are false.

Suppose that p, q, r, s, and t are all propositional variables. (a) Describe in words when the expression p ∨ q ∨ r ∨ s ∨ t is true and when it is false.

The expression is true when all the propositional variables are true. It is false when at least one of the propositional variables is false.

Suppose that p, q, r, s, and t are all propositional variables. (b) Describe in words when the expression p ∧ q ∧ r ∧ s ∧ t is true and when it is false.

a) T(Sam, Math 101) c) ∀x ∃y (y ≠ Math 101 ∧ T(x, y)) e) ∀x ∃y ∃z (x ≠ Sam → (x ≠ y ∧ T(x, y) ∧ T(x, z))

The domain for the first input variable to predicate T is a set of students at a university. The domain for the second input variable to predicate T is the set of Math classes offered at that university. The predicate T(x, y) indicates that student x has taken class y. Sam is a student at the university and Math 101 is one of the courses offered at the university. Give a logical expression for each sentence. (Use ∨, ∧, ¬, →, ≠, ∃, ∀). (a) Sam has taken Math 101. (c) Every student has taken at least one class besides Math 101. (e) Everyone besides Sam has taken at least two different math classes.

∃x ∀y (F(x, y)). Negation: ¬∃x ∀y (F(x, y)). De Morgan's Law: ∀x ∃y (¬F(x, y)). English: Everyone is an enemy of someone.

The domain for variables x and y is a group of people. The predicate F(x, y) is true if and only if x is a friend of y. For the purposes of this problem, assume that for any person x and person y, either x is a friend of y or x is an enemy of y. Therefore, ¬F(x, y) means that x is an enemy of y. Translate each statement into a logical expression. Then negate the expression by adding a negation operation to the beginning of the expression. Apply De Morgan's law until the negation operation applies directly to the predicate and then translate the logical expression back into English. (Use ∨, ∧, ¬, →, ∃, ∀). (c) Someone is a friend of everyone.

a) false c) true e) false

The domain of discourse for this problem is a group of three people who are working on a project. To make notation easier, the people are numbered 1, 2, 3. The predicate M(x, y) indicates whether x has sent an email to y, so M(2, 3) is read "Person 2 has sent an email to person 3." The table below shows the value of the predicate M(x,y) for each (x,y) pair. The truth value in row x and column y gives the truth value for M(x,y). M 1 2 3 1 T T T 2 T F T 3 T T F Indicate whether the quantified statement is true or false. Justify your answer. (a) ∀x ∀y M(x,y) (c) ∃x ∃y ¬M(x,y) (e) ∀x ∃y ¬M(x,y)

a) ∃x B(x, Sam) b) ∀y ∃x B(x, y) c) ¬∃x B(x, Nancy) d) ∀x ∃y B(x, y) e) ¬∃x (B(x, Ingrid) ∧ B(x, Dominic)) f) ∀x ((x ≠ Josephine) → B(Josephine, x)) g) ∃x ∀y (B(Nancy, x) ∧ ((x ≠ y) → ¬B(Nancy, y))) h)

The domain of discourse is the members of a chess club. The predicate B(x, y) means that person x has beaten person y at some point in time. Give a logical expression equivalent to the following English statements. You can assume that it is possible for a person to beat himself or herself. (Use ∨, ∧, ¬, →, ≠, ∃, ∀). (a) Sam has been beaten by someone. (b) Everyone has been beaten before. (c) No one has ever beaten Nancy. (d) Everyone has won at least one game. (e) No one has beaten both Ingrid and Dominic. (f) Josephine has beaten everyone else. (g) Nancy has beaten exactly one person.

a) True c) true e) false

The propositional variables, p, q, and s have the following truth assignments: p = T, q = T, s = F. Give the truth value for each proposition. (a) p ∨ ¬q (c) p ∧ (q ∨ s) (e) ¬(q ∧ p ∧ ¬s)

a) false c) true e) false g) false i) true

The tables below show the values of predicates P(x, y), Q(x, y), and S(x, y) for every possible combination of values of the variables x and y. The domain for x and y is {1, 2, 3}. P 1 2 3 1 T F T 2 T F T 3 T T F Q 1 2 3 1 F F F 2 T T T 3 T F F S 1 2 3 1 F F F 2 F F F 3 F F F Indicate whether each of the quantified statements is true or false. (a) ∃x ∀y P(x, y) (c) ∃x ∀y P(y, x) (e) ∀x ∃y Q(x, y) (g) ∀x ∀y P(x, y) (i) ∀x ∀y ¬S(x, y)

a) ∃x ∃y (x/y < 1) c) ∃x ∃y ( x + y = xy) e) ∀x ((x > 0 ∧ x < 1) → (1/x > 1)) g) ∀x ( x ≠ 0 → ((x (1/x)) = 1))

Translate each of the following English statements into logical expressions. The domain of discourse is the set of all real numbers. (Use ∨, ∧, ¬, →, ≠, ∃, ∀). (a) There are two numbers whose ratio is less than 1. (c) There are two numbers whose sum is equal to their product. (e) The reciprocal of every positive number less than one is greater than one. (g) Every number besides 0 has a multiplicative inverse.

¬∀x (¬P(x) → Q(x)) ∃x ¬(¬P(x) → Q(x)) ∃x ¬(¬¬P(x) ∨ Q(x)) ∃x ¬(P(x) ∨ Q(x)) ∃x (¬P(x) ∧ ¬Q(x))

Use De Morgan's law for quantified statements and the laws of propositional logic to show the following equivalences (Use ∨, ∧, ¬, →, ∃, ∀).: (b) ¬∀x (¬P(x) → Q(x)) ≡ ∃x (¬P(x) ∧ ¬Q(x))

¬∀x (P(x) ∧ ¬Q(x)) ∃x ¬(P(x) ∧ ¬Q(x)) ∃x (¬P(x) ∨ ¬¬Q(x)) ∃x (¬P(x) ∨ Q(x))

Use De Morgan's law for quantified statements and the laws of propositional logic to show the following equivalences (Use ∨, ∧, ¬, →, ∃, ∀): (a) ¬∀x (P(x) ∧ ¬Q(x)) ≡ ∃x (¬P(x) ∨ Q(x))

a) True, because the conclusion is true. c) True, both the hypothesis and conclusion are true.

Which of the following conditional statements are true and why? (a) If February has 30 days, then 7 is an odd number. (c) If 7 is an odd number, then February does not have 30 days.