cpsc 311 final exam

What is the output of the following circuit? (only circuit problem) (X+Y) . (Y+Z) . (X+Y+Z) (X+Y) . (Y+Z') . (X+Y+Z) (XY) + (YZ') + (XYZ) (XY) + (YZ) + (XYZ)

(XY) + (YZ') + (XYZ)

| 0 -1 7 | |-2 0 -2 | | | + | | | 0 6 2 | | -3 5 -1 |

-2 -1 5 -3 11 1

a = 1 3 -3 3 0 5 b= 3 0 -3 1 0 5 What is AB? 3 -9 0 0 0 25 -12 -6 25 9 -6 -12 9 25 AB is undefined

-6 -12 9 25

1 • 0 = 0 1

0

Use the following values for the variables: x = 0, y = 1, z = 1. What is the value for the following Boolean expression? (x+y)(z̅+̅1̅) 0 1

0

Use the following values for the variables: x = 0, y = 1, z = 1. What is the value for the following Boolean expression? (x+y̅)z 0 1

0

Let A= 1 0 0 1 0 1 0 1 1 1 1 1 The second column of matrix A is: 0 0 1 0 1 1 1 1 1

0 1 1

1 + 0 = 0 1

1

100 ∑ j j=1 What is the lower limit of the sum with index j? your answer should be a number i.e., 1

1

Define the sequence {ₐₖ}, beginning with a1=7: 7, 10, 13, 16, 19, 22, 25 what is a2?

10

Indicate which statements are propositions: 10 is a prime number. Shut the door. Would you like some cake? January has 31 days and February has 33 days.

10 is a prime number. January has 31 days and February has 33 days.

100 ∑ j j=1 What is the upper limit of the sum with index j? your answer should be a number i.e., 1

100

Consider the following pseudocode fragment: count = 0 For i = 1 to 3 For j = 1 to 4 count := count + i ⋅ j Endfor Endfor How many times is the variable count increased?(your answer should be a number i.e., 2)

12

Consider the following recurrence relation. How many initial values must be given for the sequence to be well defined for all n ≥ 0? your answer should be a number i.e., 5 cₙ=3⋅cₙ₋₁ + 2 ⋅cₙ₋₂

2

Let A= 1 0 0 1 0 1 0 1 1 1 1 1 The size of matrix A is: 4 x 3 3 x 4 12 x 1 1 x 12

3 x 4

Compute the following double sums 3. 2 ∑ ∑ (3i+2j) i=0 j=0

78

How many rows are in a truth table for a compound proposition with propositional variables p, q, and r? (your answer should be a number with no spaces e.g. 2)

8

What is the simplest form of the following boolean algebra expression? A'BC + AB'C' + AB'C + ABC' + ABC ABC + AB'C' A BC A'BC + A

A'BC + A

Let A be a 3 x 4 matrix, B be a 4 x 5 matrix, and C be a 4 x 4 matrix. Determine which of the following products are defined and find the size of those that are defined. Check all that apply AB BA AC CA BC

AB AC

Indicate whether the function defined below is one-to-one, onto, neither, or both: t: Z → Z. t(x) = x. Neither one-to-one nor onto one-to-one but not onto onto but not one-to-one Both onto and one-to-one

Both onto and one-to-one

Consider the following set:A = { 2, 3, 5, 7, 14 } 2 ∈ P(A) True False

False

Consider the following set:A = { 3, 4, { 3, 4 }, { 1, 2, 3 }, 5 } The cardinality of A is 8. True False

False

Consider the following sets: A = { 3, 4, 5 } B = { 4, 5, 3 } C = { x ∈ Z: x is odd } D = { 3, 5, 7, 9 } Is A ⊂ B ? True False

False

Consider the following sets: A = { 4, 6, 3} B = { 2, 4, 6, ..., 20 } C = { 2, 4, 6, ... } D = { 3, 4, 6 } Is 5 ∈ A ?

False

Every student loves some student. Can be translated into: ∀ x ( Student(x)→ ∃ y ( Student(y) →Loves(x,y) ))

False

Every student takes at least one course. ∀ x( Student(x)^∃ y(Course(y)^Takes(x,y))

False

For the following questions, the domain for the variable x is the set of allpositive integers (Note: the number zero is neither positive nor negative). Define the quantifiers O and M as follows: O(x): x is odd M(x): x is an integer multiple of 4 (e.g., 4, 8, 12,...) Indicate whether the following quantified statement is true or false: ∃ x (O(x) ∧ M(x))

False

There exists a smart student. Can be translated into:∃ x ( Student(x) → Smart(x) )

False

Consider a proof of the following fact: For all n ≥ 5, 2^n ≥ n^2 Using proof by Induction, what should be proven in the base case (step 1)? For n=1, 2^n ≥ n^2 For n=5, 2^n ≥ n^2 2^k ≥ k^2 If 2^k ≥ k^2 for k ≥ 5, then k^k+1 ≥ (k+1)^2

For n=5, 2^n ≥ n^2

Consider a proof of the following fact: For all n ≥ 5, 2^n ≥ n^2 Using proof by Induction, what should be proven in the base case (step 3)? For n=1, 2^n ≥ n^2 For n=5, 2^n ≥ n^2 2^k ≥ k^2 If 2^k ≥ k^2 for k ≥ 5, then k^k+1 ≥ (k+1)^2

If 2^k ≥ k^2 for k ≥ 5, then k^k+1 ≥ (k+1)^2

Is n^2=O(n)?

No

f(x) = n^5 + 4n - 3 g(x) = 1001 ⋅ n - 100 h(x) = 4 ⋅ n^2 -n +3 is f(x)=θ(n^4)?

No

Consider the function specified by the following arrow DIAGRAM(not picture) Does f have an inverse? a>x b>v c>z d>w >y Yes No

No because nothing goes to (y)

Consider the following arrow DIAGRAM(not pictured) f: A>x B>w,y C>z D>x Is f a function? No. There is an unmapped element in the domain. No. An element in the domain is mapped to two different elements. Yes.

No. An element in the domain is mapped to two different elements.

What is an upper bound on the asymptotic worst-case running time of the following algorithm: FindDuplicate This algorithm determines if there is a pair of duplicate entries in the sequence. Input: a1, a2,...,an, where n is the length of the sequence. Output: "Yes" if there is a duplicate pair and "No" if the elements in the sequence are distinct. For i = 1 to n For j=i+1 to n if (a;= a; ), then Return("Yes") End-for End-for Return("No") O(1) O(n) O(n^2)

O(n^2)

Construct a truth table to determine whether the following compound propositions are tautologies, contradictions, or neither. (p ∧ q) → p Tautology Contradiction Neither a tautology or a contradiction

Tautology

Assume propositions p, q, and r have the following truth values: p is true q is true r is false What are the truth values for the following compound proposition? p ∧ q

True

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

True

Consider the following sets: A = { 1, 2, 3 } B = { x, y } C = { u, v, w } D = { +, * } (w, y, 2) ∈ C x B x A True False

True

Consider the following sets: A = { 3, 4, 5 } B = { 4, 5, 3 } C = { x ∈ Z: x is odd } D = { 3, 5, 7, 9 } Is A ⊆ B ? True False

True

Consider the following sets: A = { 4, 6, 3} B = { 2, 4, 6, ..., 20 } C = { 2, 4, 6, ... } D = { 3, 4, 6 } Is A == D ? True False

True

For the following questions, the domain for the variable x is a group of employees working on a project. The predicate N(x) says that x is a new employee. The predicate D(x) says that x met his deadline. Consider the group defined in the table below: Name N(x) D(x) Happy F F Sleepy F T Grumpy T T Bashful T T Is the statement "Every new employee met his deadline" true or false for the group defined in the table?

True

For the following questions, the domain for the variable x is a group of employees working on a project. The predicate N(x) says that x is a new employee. The predicate D(x) says that x met his deadline. Consider the group defined in the table below: Name N(x) D(x) Happy F F Sleepy F T Grumpy T T Bashful T T The following logical expression ∀x (N(x) → D(x)) is equivalent to the statement: "Every new employee met his deadline"

True

In the following questions, the domain is the set of all positive integers (Note: the number zero is neither positive nor negative). Indicate whether the universally quantified statement is true or false. ∀x x-1 ≥ 0.

True

In the following questions, the domain is the set of all positive integers (Note: the number zero is neither positive nor negative). Indicate whether the universally quantified statement is true or false. ∃x x -1 > 0.

True

No student loves Bill. Can be translated into:¬ ∃ x ( Student(x) ∧ Loves(x, Bill) )

True

|A∪B|=|A|+|B|−|A∩B| True False

True

р | q | г | р ^ (not)q | (р^q) vг T T T F (a) T T F F F T F T T T T F F (b) T What is the correct value for A?

True

f(x) = n^5 + 4n - 3 g(x) = 1001 ⋅ n - 100 h(x) = 4 ⋅ n^2 -n +3 is f(x)=Ω(n^4)?

Yes

f(x) = n^5 + 4n - 3 g(x) = 1001 ⋅ n - 100 h(x) = 4 ⋅ n^2 -n +3 is g(x)=θ(n)?

Yes

let f(n) = n^4 -3n +4. is f(n)=O(n^4)?

Yes

xy + x(has - above)y + zy is equivalent to y hint (use the laws of Boolean algebra) Yes No

Yes

Put the steps in the correct order to show that ¬(p → q) ≡ p ∧ ¬q. Note: start with the left-hand side ____p ∧ ¬q ____¬(¬p ∨ q) ____¬¬ p ∧ ¬q

__3__p ∧ ¬q __1__¬(¬p ∨ q) __2__¬¬ p ∧ ¬q

Not all roses are red can be translated to FOL as: ∃x (rose(x) ∧ ¬ red(x) ) ¬ ∀ x(rose(x) →¬ red(x) )

both

Indicate whether the function defined below is one-to-one, onto, neither, or both: t: Z → Z. t(x) = 2x. Neither one-to-one nor onto one-to-one but not onto onto but not one-to-one

one-to-one but not onto

This question uses the following propositions: p: I will share my cookie with you. q: You will share your soda with me. Select the conditional statement that has the same logical meaning as the English sentence given. If you share your soda with me, then I will share my cookie with you.

q → p

Consider the following sets: A = { 1, 2, 4, 11, 12 } B = { 2, 4, 7, 8, 11 } C = { x ∈ Z: x is even } D = { x ∈ Z: x is odd } ( A ∩ D ) ∪ B = { 1, 2, 4, 7, 8, 11 } { 2, 4, 7, 8, 11 } { 2, 4 }

{ 1, 2, 4, 7, 8, 11 }

Consider the following sets:A = { 1, 2, 4, 11, 12 } B = { 2, 4, 7, 8, 11 } A ∩ B = {1, 2, 4, 7, 8, 11, 12 } { 7, 11 } { 2, 4, 11 }

{ 2, 4, 11 }

Consider the following arrow DIAGRAM(not picture) g: a>z b>z c>w d>x What is the range of g? {a, b, c, d} {w, x, y, z} {w, x, z}

{w, x, z}

Let ₛₙ be the salary of an employee in year n. The employee receives a $1000 bonus in December. The salary for the next year is 5% more than the income received by the employee in the previous year(including salary and bonus) ₛₙ = 1.05 ⋅ (ₛₙ₋₁) + 1000 ₛₙ = 1.5 ⋅ (ₛₙ₋₁ + 1000) ₛₙ = 1.05 ⋅ (ₛₙ₋₁ + 1000)

ₛₙ = 1.05 ⋅ (ₛₙ₋₁ + 1000)