math foundations

Assume that x, y, and z are all Boolean variables. Which one of the following Boolean expressions is in Conjunctive Normal Form?

left parenthesis x plus y with bar on top plus z right parenthesis left parenthesis x with bar on top plus z right parenthesis (x+!y+z)(!x+z)

Which of the following correctly represents the proposition that there exist a positive number x, where for every negative number y the sum of x and y is positive? Assume the domain of discourse is the set of integers.

∃x∀y ((x>0 )∧ ((y<0) → (x+y>0))

Assume that the domain of discourse is the set of employees at a company. Define the predicate: N(x, y): x earns more than y Select the logical expression that is equivalent to: "Some employee of the company earns more than everyone else who works there.

∃x∀y (x≠y → N(x,y))

Which one of the following expressions has the largest value among provided answers?

⌈−0.9⌉

Consider the following circuit. Which Boolean function does the output Y represents?

!(x+y) + y * z

Which of the following does NOT belong to the set {|x - y| | x ∊ {-1, 2, -3}, y ∊ {2 , -3, 4}}?

1

There are 10 boys and 10 girls in a class. Answer questions: 1) We need to select 3 students to participate an event. How many possible selections? 2) We need to select 3 students to participate an event. But at least one girl, how many possible selections? 3) We need to select 3 students for 3 different prizes, how many possible selections?

1) 20 choose 3, i.e., C(20,3) =1140 2) At least one girl, means 1 girl, 2 girls or 3 girls. 1 girl: 10*C(10,2)= 450 2 girls: C(10,2) *10 = 450 3 girls: C(10,3) = 120 Total: 1020 3) P(20,3) = 6840

1) What is the total degree of this graph? 2) What are the adjacent vertices of vertex E?

1.20 2. a,b,g,f

Consider the follow theorem: If x is a positive integer less than 4, then (x + 1)3 ≥ 4x. Which set of facts must be proven in a proof by exhaustion of the theorem?

23 ≥ 41 33 ≥ 42 43 ≥ 43

Which of the following is a predicate?

2k is an odd number

License plate numbers in a certain state consists of seven characters. The first character is a digit (0 through 9). The next four characters are capital letters (A through Z) and the last two characters are digits. Therefore, a license plate number in this state can be any string of the form: Digit-Letter-Letter-Letter-Letter-Digit-Digit How many different license plate numbers are possible?

456,976,000 10^3 * 26^4

Theorem: For any real number x, if 0 ≤ x ≤ 3, then 15 - 8x + x2 > 0 Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem?

Assumed: 15 - 8x + x^2 ≤ 0 Proven: x < 0 or x > 3

Theorem: For any two real numbers, x and y, if x and y are both rational then x + y is also rational. Which facts are assumed and which facts are proven in a direct proof of the theorem?

Assumed: x is rational and y is rational. Proven: x + y is rational

Let F(x,y) be the statement "x is a friend of y". Which of the following does the proposition ∀x∃y ¬F(x,y) represent? Assume the domain of discourse is the set of all people.

Everybody is not a friend of somebody.

Let P(x) denote "x has a bachelor's degree in CIS" and Q(x) denote "x is an information systems analyst". The domain of discourse is the set of all people. Which of the following propositions does ∀x (P(x)→Q(x)) represent?

Everyone who has a bachelor's degree in CIS is an information systems analyst.

Let P(x) denote "x is a professional athlete" and Q(x) denote "x plays soccer". The domain of discourse is the set of all people. Which of the following correctly describes ∀x (P(x) ∨ ¬Q(x))?

Everyone who plays soccer is a professional athlete.

Assume that proposition p is true and the proposition q and r are false. What is the truth value of the proposition (p ∨¬r)↔ (¬q →r)?

False

Assume that x, y and z are Boolean variables. Then the Boolean expressions !(x * y * z) and !x * !y * !z are equivalent.

False

Let p,q and r be three propositions. Then (p→q)→r ≡ p→(q→r).

False

Let P(x) denote "x visited Brownsville" and Q(x) denote "x visited South Padre Island". The domain of discourse is the set of all people. Which of the following correctly describes ∃x (P(x)→Q(x))?

For some person, if he/she visited Brownsville, then he/she also visited South Padre Island.

Select the sentence that is equivalent to the following statement: Among any two consecutive positive integers, there is at least one integer that is not prime.

If x is a positive integer, then x is not prime or x + 1 is not prime.

Define p to be the proposition that it is snowing and q to be the proposition that it is warm. Which of the following propositions does q→¬p represent?

It does not snow when it is warm

Let set A be the set of multiples of 3 and set B be the set of odd integers. What is A - B?

Set of multiples of 6.

Assume that x, y and z are Boolean variables. Then the value of the Boolean expression (x + !y + z) (!(Y+!Z) + !x) with x=1, y=0, and z=1 is

True

Assume the domain of discourse is the set of positive integers. Then the quantified statement Ex (x > 1) ^ (x^2-2x lessthanorequal to 0) is true.

True

Let f: X → Y be a function that maps elements of a set X (domain) to those of a set Y (target). Which of the following statements is possibly false?

c. For every y ϵ Y, there is a unique x ϵ X such that f(x) = y.

Consider the function f: R→R, where R represents the set of all real numbers and for every x ϵ R, f(x) = x^3. Which of the following statements is true?

f is one-to-one and onto.

Consider the function g: Z→ {0, 1, 2, 3, 4}, where Z represents the set of all integers and for every x ϵ Z, g(x) = x mod 5, i.e. g(x) is the remainder of x divided by 5. Which of the following statements is true?

g is onto but not one-to-one.

Define p to be the proposition that it is snowing and q to be the proposition that it is warm. Which of the following denotes the proposition that snowing is a sufficient condition for it to be not warm?

p→¬q

Theorem: If r and s are rational numbers, then the product of r and s is a rational number. Which facts are assumed in a direct proof of the theorem

r = a/b, and s = c/d, where a, b, c, d are integers and b ≠ 0 and d ≠ 0.

Consider the following logical proof of ∃y (¬Q(y)) from the hypotheses that ∀x∃y (Q(y) →¬ P(x)) and ∃x (P(x)). ∃x (P(x)) P(c) for some c Rule (1) ∀x∃y (Q(y) → ¬P(x)) ∃y (Q(y) → ¬P(c)) for c Rule (2) Q(d) → ¬P(c)) for c and some d Rule (3) ¬Q(d) for some d Rule (4) ∃y (¬Q(y)) Rule (5) Match each of the above rules (1)-(5) with one of the inference rules in Table 1.11.1 of the zyBook for this course.

rule 1: A. Ecistential Instantiation rule 2: E. Universal instantiation rule 3: A. Existential Instantiation rule 4: I. Modus Tollens rule 5: D. Existential Generalization

Consider the following input/output table for a Boolean function f. Which of the Boolean expressions provided below represents f in Disjunctive Normal Form?

x * y * z + x * !y * !z + !x * y * !z + !x * !y * z

Which of the following statements about the floor and ceiling functions can be true for some real number x?

x - ⌊x - 0.5⌋ ≥ 1

Theorem: For any two real numbers, x and y, such that y ≠ 0, |x/y| = |x|/|y|. One of the cases in the proof of the theorem says: Since |x| = x and |y| = -y, |x|/|y| = x/(-y) = -x/y = |x/y|. Select the case that corresponds to this argument.

x ≥ 0 and y < 0

Assume that x, y and z are Boolean variables. Which of the following Boolean expressions is equivalent to left parenthesis x plus y plus z right parenthesis left parenthesis x plus y plus z with bar on top right parenthesis ?

x+y

Which of the following is a counter-example to the statement below? For every integer x, x < x^2.

x= 1

Let f(x, y, z) = x y+z be a Boolean function on three Boolean variables, x, y and z. Then the domain and range of f are

{0,1}^3 and {0,1}

Which of the following forms a partition of R?

{x ∈ R: x < 2} {x ∈ R: 2 ≤ x < 4} {x ∈ R: 4 ≤ x}

Define set A = {a, b} and set B = {1, 2, 3}. Which of the following is FALSE?

|A × B| = 8

Assume the domain is the set of all positive integers. Which of the following quantified statements is false?

∀x (2x > x +1)

Assume that the domain of discourse 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) = T because the value in row 2, column 3, is T.

∀x ∀y ((x ≠ y) → P(x, y))

Let P(x) and Q(y) be two predicates. Then ∀x(¬ (∃y(P(x) ∧ ¬Q(y))) is logically equivalent to

∀x∀y (¬ P(x) ∨ Q(y))

Let P(x) and Q(x) be two predicates. Then ¬∀x (P(x)→Q(x)) is equivalent to

∃x (P(x)∧ ¬Q(x))

Let P(x) denote "x has a bachelor's degree in CIS" and Q(x) denote "x is an information systems analyst". The domain of discourse is the set of all people. Which of the following correctly represents the proposition that someone who has a bachelor's degree in CIS is an information systems analyst?

∃x (P(x)∧Q(x))

Let P(x) denote "x visited Brownsville" and Q(x) denote "x visited South Padre Island". The domain of discourse is the set of all people. Which of the following correctly represents the proposition that not everyone who visited Brownsville also visited South Padre Island?

∃x (P(x)∧¬Q(x))

Let P(x) denote "x is a professional athlete" and Q(x) denote "x plays soccer". The domain of discourse is the set of all people. Which of the following correctly represents the proposition that someone who plays soccer is not a professional athlete.

∃x (¬P(x) ∧ Q(x))

Assume the domain of discourse 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."

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

Which of the following can be considered as a predicate?

All of the answers provided.