Midterm Quiz 1
The propositional variables b, v, and s represent the propositions: b: Alice rode her bike today. v: Alice overslept today. s: It is sunny today. Select the logical expression that represents the statement: "Alice rode her bike today only if it was sunny today and she did not oversleep". a) b → (s → ¬v) b) (s ∧ ¬v) → b c) b → (s ∧ ¬v) d) s ∧ (¬v → b)
c) b → (s ∧ ¬v)
p = F, q = T, and r = T. Select the expression that evaluates to false. a) q ^ r b) q V r c) ¬q d) p V r
c) ¬q
Select the statement that is not a proposition. a) 5 + 4 = 8 b) It will be sunny tomorrow. c) Chocolate is the best flavor. d) Take out the trash.
d) Take out the trash.
Determine whether the following pairs of expressions are logically equivalent. 1. ¬(p ∨ ¬q) and ¬p ∧ q 2. ¬(p ∨ ¬q) and ¬p ∧ ¬q 3. p ∧ (p → q) and p → q 4. p ∧ (p → q) and p ∧ q
1. Logically 2. Not logically 3. Not logically 4. Logically
The domain of discourse is the set of all positive integers. P(x): x is even T(x, y): 2x = y E(x, y, z): xy = z Find whether each logical expression is a proposition. If the expression is a proposition, then determine its truth value. 1. P(3) 2. ¬P(3) 3. T(5, 32) 4. T(5, x) 5. E(6, 2, 36) 6. E(2, y, 7) 7. P(3) ∨ T(5, 32) 8. T(5, 16) → E(6, 3, 36)
1. Proposition. Truth value is false. 2. Proposition. Truth value is true. 3. Proposition. Truth value is true. 4. Not a proposition. 5. Proposition. Truth value is true. 6. Not a proposition 7. Proposition. Truth value is true. 8. Proposition. Truth value is 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 Match the TRUE or FALSE values for the following compound propositions. 1. ¬p 2. p ∨ r 3. q ∧ s 4. q ∨ s 5. q ⊕ s 6. q ⊕ r
1. True 2. False 3. True 4. True 5. False 6. True
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. 1. p ∨ ¬q 2. (p ∧ q) ∨ s 3. p ∧ (q ∨ s) 4. p ∧ ¬(q ∨ s) 5. ¬(q ∧ p ∧ ¬s) 6. ¬(p ∧ ¬(q ∧ s))
1. True 2. True 3. True 4. False 5. False 6. False
Match each English sentence with the correct logical expression using the propositional variables defined below. p: the applicant has written permission from his parents e: the applicant is at least 18 years old s: the applicant is at least 16 years old 1. The applicant has written permission from his parents and is at least 16 years old. 2. The applicant has written permission from his parents or is at least 18 years old. 3. The applicant does not have written permission from his parents or is not at least 16 years old. 4. The applicant does not have written permission from his parents and is not at least 18 years old.
1. p ^ s 2. p ∨ e 3. ¬p ∨ ¬s 4.¬p ∨ ¬e
Match the following English definitions with correct propositions. c: I will return to college. j: I will get a job. 1. Not getting a job is a sufficient condition for me to return to college. 2. If I return to college, then I won't get a job. 3. I am not getting a job, but I am still not returning to college. 4. There's no way I am returning to college. 5. I will get a job and return to college.
1. ¬j → c 2. c → ¬ j 3. ¬c → ¬ j 4. ¬c 5. j ^ c
Which of the following sentences are propositions? (Multiple answers) 1. Do you like my new shoes? 2. It's a beautiful day. 3. There is a number that is larger than 17 4. Have a nice day. 5. Every prime number is even 6. The patient has diabetes. 7. The light is on. 8. The soup is cold. 9. 2 + 3 = 6 10. The sky is purple.
2,3,5,6,7,8,9,10
Use De Morgan's law to select the statement that is equivalent to: "It is not true that the employee received a large bonus and has a big office." a) The employee did not receive a big bonus or does not have a big office. b) The employee received a big bonus and has a big office. c) The employee received a big bonus or has a big office. d) The employee did not receive a big bonus and does not have a big office.
a) The employee did not receive a big bonus or does not have a big office.
The domain for variable x is the set of all integers. Select the statement that is false. a) ∀x (x^2 > x) b) ∀x (x^2 ≥ x) c) ∃x (x= x) d) ∀x (x^2 ≠ 5)
a) ∀x (x2 > x)
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(4, 0 2) d) T(1, 1, 1)
b) T(4, 1, 25)
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 (P(x) ∨ Q(x)) b) ∀x (Q(x) → P(x)) c) ∀x (P(x) → Q(x)) d) ∀x (P(x) ∧ Q(x))
b) ∀x (Q(x) → P(x))
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 (S(x) ↔ A(x)) b) ∀x (S(x) → A(x)) c) ∀x (A(x) → S(x)) d) ∀x (S(x) ∧ A(x))
b) ∀x (S(x) → A(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))
b) ∃x (P(x) ∧ ¬Q(x))
Select the law which shows that the two propositions are logically equivalent. r ∧ (p ∨ q) r ∧ (q ∨ p) a) Distributive law b) Associative law c) DeMorgan'€™s law d) Commutative law
d) Commutative law
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) At least one student was not absent yesterday. b) Every student was absent yesterday. c) Some student was absent yesterday. d) Every student was not absent yesterday.
d) Every student was not absent yesterday.
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))
d) ∃x (¬P(x) ∧ Q(x))
