CS 113 Study
Which of these is an example of Modus Tollens? a) If Sally had side effects, then she took the medication. Sally did not take the medication. Therefore, she did not have side effects. b) If Sally took the medication, then she had side effects. Sally did not take the medication. Therefore, she did not have side effects. c) Both d) None
a) If Sally had side effects, then she took the medication. Sally did not take the medication. Therefore, she did not have side effects.
Given this function f, what is the domain and the range of f? f = {(a,2), (b,3), (c,4), (d,2)} a) domain = {a, b, c, d}, range = {2, 3, 4} b) domain = {2, 3, 4}, range = {a, b, c, d} c) domain = {a, b, c, d}, range = {1, 2, 3, 4} d) None
a) domain = {a, b, c, d}, range = {2, 3, 4}
Which of the following is a linear homogenous recurrence relation? a) f_n = f_(n-1) - f_(n-3) b) f_n = n * f_(n-1) - f_(n-3) c) Both d) None
a) fn = fn-1 - fn-3
Which of the following is true, if p = T, q = T, and r = F? a) p ∨ q → ¬r b) p ∧ q → r c) Both d) None
a) p ∨ q → ¬r
Is this argument valid or invalid p q ------- * p ⟷ q a) valid b) invalid c) More information is needed d) None
a) valid
Is this argument valid or invalid? p ⟷ q p ∨ q ------- * p a) valid b) invalid c) More information needed d) None
a) valid
A = {1,2,3,4}. P(A) is a power set of A. Select the statement that is false a) {2,3} ⊆ P(A) b) Ø ⊆ P(A) c) Ø ∈ P(A) d) None
a) {2,3} ⊆ P(A)
Use De Morgan's law to select the statement that is logically equivalent to: ∃x ∃y (¬P(x) ∨ ¬Q(y)) a) ¬ ∀x ∀y (P(x) ∧ Q(y)) b) ∀x ∀y (P(x) ∧ Q(y)) c) Both d) None
a) ¬ ∀x ∀y (P(x) ∧ Q(y))
Which of the following evaluates to true, if p = T, q = F, and r = T? a) ¬(q ∧ r) → p b) (p ∧ r) → q c) Both d) None
a) ¬(q ∧ r) → p
The domain is the set of all real numbers. Which statement is equivalent to "There is no smallest number"? a) ¬∃x ∀y (x ≤ y) b) ¬∃x ∃y (x ≤ y) c) Both d) None
a) ¬∃x ∀y (x ≤ y)
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 (A(x) → S(x)) c) Both d) None
a) ∀x (S(x) → A(x))
Select the statement that is true a) ∀x ∃y (x + y = 0) b) ∃x ∀y (x + y = 0) c) Both d) None
a) ∀x ∃y (x + y = 0)
Select the law which shows that the two propositions are logically equivalent. ¬(p ∨ q) ∨ (¬p ∧ q) (¬p ∧ ¬q) ∨ (¬p ∧ q) a) Commutative law b) DeMorgan's law c) Distributive law d) Associative law
b) DeMorgan's law
The domain of the relation R is {a,b}. R = {(a,b), (b,a), (a,a), (b,b)}. The relation is a) Anti-symmetric b) Symmetric c) Neither
b) Symmetric
What is the common ratio of the following geometric sequence: 27, 9, 3, 1, ... a) 9 b) 3 c) 1/3 d) None
c) 1/3
Which of the following statements is true? a) If 3 is an odd number, then 9 is an odd number. b) If 4 is an odd number, then 9 is an odd number. c) Both d) None
c) Both
Select that law that establishes that the two sets are equal. A ∩ ~(B U C) = A ∩ (~B ∩ ~C) a) Idempotent law b) Absorption law c) De Morgan's law d) None
c) De Morgan's law
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 generalization b) Existential instantiation c) Existential generalization d) None
c) Existential generalization
A function f: X → Y is __________ if x_1 ≠ x_2 implies that f(x_1) ≠ f(x_2). That is, f maps different elements in X to different elements in Y. a) Bijective b) Surjective c) Injective d) None
c) Injective
Select the set that is equal to 3,5,7,9,11,13 a) {x ∈ Z: 3 < x < 14} b) {x ∈ R: 3 ≤ x < 14} c) {x ∈ Z: x is odd and 3 ≤ x < 14} d) None
c) {x ∈ Z: x is odd and 3 ≤ x < 14}
The domain for variable x is the set of all integers. Select the statement that is true. a) ∀x (x^2 = 1) b) ∃x (3x = 1) c) ∃x (x^2 < 1) d) None
c) ∃x (x^2 < 1)
Select the logical expression that is equivalent to: S(x): x studied for the test A(x): x received an A on the test "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) None
c) ∃x (¬S(x) ∧ A(x))
What is the coefficient of the c^4 d^7 term in (-3c + 5d)^11? a) (11 choose 7) x (3^4) x (5^7) b) (-3^4) x (5^7) c) (11 choose 7) d) None
a) (11 choose 7) * (3^4) * (5^7)
Which of the following evaluates to true, if p = F, q = T, and r = T? a) (¬p ∧ r) → q b) (q ∨ ¬r) → p c) Both d) None
a) (¬p ∧ r) → q
Which statement is the contrapositive of: "If x=4 , then 3x=12 ." a) 3x≠12 , then x≠4 . b) If x=4 , then 3x=12 . c) If n≠12 , then 3x≠12 . d) If 3x=12 , then x=4 .
a) 3x≠12 , then x≠4 .
Given the set A = {1, 2, {3,4}, {5,6,7}}. Select the correct value for |A| a) 4 b) 5 c) 6 d) None
a) 4
How many strings of length 10 over the alphabet {a,b,c,d} have at least one b somewhere in the string? a) 4^10 - 3^10 b) 3^10 c) 10 * 4^9 d) None
a) 4^10 - 3^10
Select the set that is equivalent to C U (C ∩ B). a) C b) B ∩ C c) Ø d) None
a) C
Select the law which shows that the two propositions are logically equivalent. p ∨ ¬q ¬q ∨ p a) Commutative law b) DeMorgan's law c) Distributive law d) Associative law
a) Commutative law
Select the law that establishes that the two sets are equal ((A ⊕ B) - C) ∩ Ø = Ø a) Domination law b) Idempotent law c) De Morgan's law d) None
a) Domination law
Select the law that established that the two sets are equal. (A ∩ B) U (A ∩ B) = A ∩ B a) Idempotent law b) Absorption law c) De Morgan's law d) None
a) Idempotent law
Select the statement that is false. a) If 3 is a prime number, then 6 is a prime number. b) If 4 is a prime number, then 5 is a prime number. c) If 3 is a prime number, then 5 is a prime number. d) If 4 is a prime number, then 6 is a prime number.
a) If 3 is a prime number, then 6 is a prime number.
Select the statement that is true. a) If 7 is an odd number, then February does not have 30 days. b) If January has 31 days, then 7 is an even number. c) Both d) None
a) If 7 is an odd number, then February does not have 30 days.
Select the description that fits the sequence: 5, 5, 5, 5 a) Non-increasing and non-decreasing b) Non-decreasing and increasing c) Non-increasing and decreasing d) None
a) Non-increasing and non-decreasing
Select the description that fits the sequence 8,5,2,2,1,-1 a) Non-increasing but not decreasing b) Non-decreasing but not increasing c) Non-increasing and decreasing d) None
a) Non-increasing but not decreasing
Function f:{0,1}^3 -> {0,1}^3. f(x) is obtained by removing the second bit from x and placing the bit at the end of the string. For example, f(101) = 101. Select the correct description of the function f. a) One-to-one and onto b) One-to-one but not onto c) Onto but not one-to-one d) Neither onto nor one-to-one
a) One-to-one and onto
A function f: X → Y is __________ if the range of f is equal to the target Y. That is, for every y ∈ Y, there is an x ∈ X such that f(x) = y. a) Surjective b) Bijective c) Injective d) None
a) Surjective
Another word for Onto is a) Surjective b) Injective c) Bijective d) None
a) Surjective
The predicate T is defined as: T(x,y,z): (x+y)^2 = z Select the proposition that is true. a) T(4, 1, 25) b) T(4, 1, 5) c) T(1, 1, 1) d) T(4, 0 2)
a) T(4, 1, 25)
The domain of the relation R is {a,b}. R = {(a,b), (b,a), (a,a), (b,b)}. The relation is a) Transitive b) Not transitive c) Neither
a) Transitive
Consider the following propositions. p: 8 < 5 q: 3 < 9 What is the truth value of the negation of p? a) True b) False c) Both d) Cannot be determined with given information
a) True
Which collection of sets forms a partition of: {1,2,3,4,5,6,7,8}? a) {1, 2, 5, 7}, {3, 4, 6}, {8} b) {1, 2, 5, 7}, {3, 4, 8}, {8} c) Neither d) None
a) {1, 2, 5, 7}, {3, 4, 6}, {8}
The domain for variable x is the set of all integers. Select the statement that is true. a) ∀x (x^2 ≠ 5) b) ∀x (x^2 > x) c) Both d) None
a) ∀x (x^2 ≠ 5)
The domain for the first input variable to predict T is a set of students at a university. The domain for the second input variable to predict T is the set of Math classes offered at that university. The predicate T(x,y) indicates that student x has taken class y. Which of the following is equivalent to "Every student has taken at least one math class"? a) ∀x ∃y T(x,y) b) ∃x ∀y T(x,y) c) Both d) None
a) ∀x ∃y T(x,y)
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 (¬P(x) ∨ Q(x)) a) ∃x (P(x) ∧ ¬Q(x)) b) ∀x (P(x) ∧ ¬Q(x)) c) ∃x (¬P(x) ∧ Q(x)) d) None
a) ∃x (P(x) ∧ ¬Q(x))
20 applicants from a pool of 90 application will be hired. How many ways are there to select the applicants who will be hired? a) P(90,20) b) (90 choose 20) c) 90^20 d) None
b) (90 choose 20)
A = {a,b}. B = {1,2,3}. Which of these is an element of A x B x B? a) (a,a,1) b) (b,2,3) c) (b,1,a) d) None
b) (b,2,3)
p = T, q = F, and r = T. Select the expression that evaluates to false. a) (q ∧ r) → ¬p b) (p ∧ r) → q c) ¬(q ∧ r) → p d) (q ∧ r) → p
b) (p∧r)→q
The sequence {f_n} starts with an index of 1 and is defined so that f_n is the largest integer k such that k^2 ≤ n. Which sequence fits the definition of {f_n} a) 1, 4, 9, 16, 25, ... b) 1, 1, 1, 2, 2, ... c) 2, 4, 8, 16, 32, ... d) None
b) 1, 1, 1, 2, 2, ...
Functions f(x) = x^2, g(x) = 2^x, h(x) = ⌈x/5⌉. What is (g o h o f)(4) a) 4 b) 16 c) 20 d) None
b) 16
Consider the relation R = {(1,4), (3,1), (1,3), (2,2), (3,4), (2,1), (4,2), (2,3)}. What is the in-degree of vertex 2? a) 1 b) 2 c) 3 d) 4
b) 2
A group consists of 10 kids and 2 adults. On a hike, they must form a line with an adult at at the front and an adult at the back. How many ways are there to from the line? a) 2 * 11! b) 2 * 10! c) 12! d) None
b) 2 * 10!
S = {a,b,c,d,e,f,g}. The power set of S, P(S), is the set of all subsets of S. What is |P(S)| ? a) 7 b) 2^7 c) 7! d) None
b) 2^7
A bank PIN is a string of four digits, each digit 0-9. How many choices are there for a PIN if the last digit must be odd and all the digits must be different from each other? a) 10 x 9 x 8 x 5 b) 9 x 8 x 7 x 5 c) Both d) None
b) 9 x 8 x 7 x 5
Select the law that shows that the two propositions are logically equivalent: r ∧ (p ∨ q); r ∧ (q ∨ p) a) DeMorgan's Law b) Commutative Law c) More information is needed d) None
b) Commutative Law
Select the law that shows that the two propositions are logically equivalent. ¬(¬p ∨ q) ∨ (p ∧ q) (¬¬p ∧ ¬q) ∨ (p ∧ q) a) Distributive law b) DeMorgan's law c) Associative law d) None of the answers are correct.
b) DeMorgan's law
Select the law that shows that the two propositions are logically equivalent. (p ∧ ¬q) ∨ (p ∧ q) p ∧ (¬q ∨ q) a) None of the answers are correct. b) Distributive law c) DeMorgan's law d) Commutative law
b) Distributive 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 not absent yesterday c) Every student was absent yesterday d) None
b) Every student was not absent yesterday
Which of the following is true about composition of functions? a) Given f and g are two functions; (f o g) = (g o f) b) Given f, g, and h, are three functions; f o g o h = (f o g) o h = f o (g o h) c) Both d) None
b) Given f, g, and h, are three functions; f o g o h = (f o g) o h = f o (g o h)
Select the statement that is not a proposition. a) The earth is flat b) How are you? c) Both d) None
b) How are you?
Which of the following is NOT a proposition? a) Chocolate is the best food. b) How are you? c) Both d) None
b) How are you?
Which of the following statements is the contrapositive for "If x = 6, then 2x = 12"? a) If 2x = 12, then x = 6 b) If 2x ≠ 12, then x ≠ 6 c) If x = 6, then 2x = 12 d) None
b) If 2x ≠ 12, then x ≠ 6
Another word for One-to-One is a) Surjective b) Injective c) Bijective d) None
b) Injective
Which of the following is NOT a proposition? a) 27 + 5 = 18 b) Keep masks on when indoors. c) Both d) None
b) Keep masks on when indoors.
Given the function h: Z -> Z. h(x) = x^3. The function h is a) Onto b) One-to-one c) Both d) None
b) One-to-one
The domain of the relation R is {a,b}. R = {(a,b), (b,a), (a,a), (b,b)}. The relation is a) Anti-reflexive b) Reflexive c) Neither
b) Reflexive
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) Someone did not get an A on the test c) Every student got an A on the test d) None
b) Someone did not get an A on the test
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 and does not have a big office. b) The employee did not receive a big bonus or does not have a big office. c) Both d) None
b) The employee did not receive a big bonus or does not have a big office.
Use De Morgan's law to select the statement that is equivalent to: "It is not true that the patient has high blood pressure or influenza." a) The patient has high blood pressure or has influenza. b) The patient does not have high blood pressure and does not have influenza. c) The patient has high blood pressure and has influenza. d) The patient does not have high blood pressure or does not have influenza.
b) The patient does not have high blood pressure and does not have influenza.
Which of the following is NOT a proposition? a) 7 is an odd number b) The value x is an odd number (x is a variable) c) Both d) None
b) The value x is an odd number (x is a variable)
Select that statement that is false a) Z ⊂ R b) Z ⊂ R+ c) Z ⊆ R d) None
b) Z ⊂ R+
Sets A = {a,b,c,d}, X = {1,2,3,4}. Select the definition for f that is a bijection. a) f = {(a,2), (b,3), (c,1), (d,1)} b) f = {(a,2), (b,3), (c,4), (d,1)} c) Both d) None
b) f = {(a,2), (b,3), (c,4), (d,1)}
Is this argument valid or invalid p ∨ q p ------- * q a) valid b) invalid c) More information is needed d) None
b) invalid
Which of the following evaluates to true, if p = T, q = F, and r = F? a) p ∧ q b) p ∨ r c) Both d) None
b) p ∨ r
The domain is the set of all integers. In the expression P(5), 5 is a a) arbitrary element b) particular element c) Both d) None
b) particular element
Select the proposition that is logically equivalent to ¬p→q . a) ¬p∧q b) p∨q c) p∧¬q d) ¬p∨q
b) p∨q
Select the converse of p → q a) p → q b) q → p c) ¬q → ¬p d) None of the answers are correct.
b) q → p
Which of the following statements is the converse for p → q? a) ¬p → ¬q b) q → p c) Both d) None
b) q → p
Select the set that corresponds to the relation given in the matrix. Rows of the matrix are numbered 1 through 4 from top to bottom and columns are numbered 1 through 4 from left to right. 0 1 0 0 0 1 0 1 0 0 1 0 0 0 0 0 a) {(1,2), (2,3), (2,4), (3,3)} b) {(1,2), (2,2), (2,4), (3,3)} c) Both d) None
b) {(1,2), (2,2), (2,4), (3,3)}
Select the 5-subset from {1, 2, 3, 4, 5, 6, 7, 8, 9} that is the next one in lexicographic order after {2, 3, 7, 8, 9}. a) {2,4,5,8,9} b) {2,4,5,6,7} c) {2,4,7,8,9} d) None
b) {2,4,5,6,7}
Sets C = {3,5,9,12,15,16}, D = {5,7,8,12,13,15}. The universal set U is the set of all integers. Select the set corresponding to C ⊕ D. a) {3,9,16} b) {3,7,8,9,13,16} c) {3,5,7,8,9,12,13,15,16} d) None
b) {3,7,8,9,13,16}
Which of the following statements is the contrapositive for p → q? a) ¬p → ¬q b) ¬q → ¬p c) q → p d) None
b) ¬q → ¬p
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 (xy ≥ 0) c) Both d) None
b) ∀x ∃y (xy ≥ 0)
A donut store sells packages of 12 donuts. The store has made x donuts. How many complete packages does the store have for sale? a) ⌊12x⌋ b) ⌊x/12⌋ c) ⌈x/12⌉ d) None
b) ⌊x/12⌋
Select the permutation that is the next one in lexicographic order after (4, 6, 2, 5, 3, 1). a) (4, 6, 2, 1, 3, 5) b) (4, 6, 5, 1, 2, 5) c) (4, 6, 3, 1, 2, 5) d) None
c) (4, 6, 3, 1, 2, 5)
A sequence is defined by the recurrence relation Fn = F(n-1) - F(n-3). How many initial values are required so that the sequence is well defined for all n ≥ 0? a) 1 b) 2 c) 3 d) None
c) 3
Consider the relation R = {(1,4), (3,1), (1,3), (2,2), (3,4), (2,1), (4,2), (2,3)}. What is the out-degree of vertex 2? a) 1 b) 2 c) 3 d) 4
c) 3
Another word for both One-to-One and Onto is a) Surjective b) Injective c) Bijective d) None
c) Bijective
The predicate T is defined as: S(x,y) : x * y = 20 Select the proposition that is true. a) S(5,4) b) S(2,10) c) Both d) None
c) Both
Which of the following evaluates to true, if p = F, q = T, and r = T? a) q ∨ r b) p ∨ r c) Both d) None
c) Both
Which of the following evaluates to true, if p = T, q = F, and r = T? a) p ∨ ¬q b) ¬(p ∧ q ∧ r) c) Both d) None
c) Both
Which of the following is true? a) A function f has an inverse if an only if f is a bijection b) A bijection is a function that is one-to-one and onto. c) Both d) None
c) Both
A coin is flipped 5 times. Each outcome is written as a string of length 5 from {H, T}, such as THHTH. Select the set corresponding to the event that exactly one of the five flips comes up heads. a) { HTTTT, THTTT, TTHTT, TTTHT } b) { HTTTT, THTTT, TTTHT, TTTTH, TTTTT } c) { HTTTT, THTTT, TTHTT, TTTHT, TTTTH } d) None
c) { HTTTT, THTTT, TTHTT, TTTHT, TTTTH }
Sets B = {x ∈ Z: x is a prime number}, C = {3,5,9,12,15,16}. The universal set U is the set of all integers. Select the set corresponding to ¬B ∩ C a) {9,12,16} b) {3,5,9,15} c) {9,12,15,16} d) None
c) {9,12,15,16}
Consider the relation: { (A,3), (B,1), (B,4), (D,3), (D,4) } Which statement below describes the relation? a) This relation is a well-formed function b) This relation is a bijection c) Both d) None
d) None
Which of the following evaluates to true, if p = F, q = T, and r = T? a) p ∨ ¬q ∨ ¬r b) ¬(q ∨ r) c) Both d) None
d) None
Which of the following values would make this argument valid? ¬q p ∨ q ------- * p ⟷ q a) p = T, q = T b) p = F, q = F c) Both d) None
d) None
What is the truth value of the following statement? What is the truth value of its inverse? If 7 < 5, then 5 < 3. a) The above statement is false and its inverse is true. b) None of the answers are correct. c) The above statement is true and its inverse is false. d) The above statement is true and its inverse is true.
d) The above statement is true and its inverse is true.
