Use Newton-Raphson to find the real root to five significant figures 64x^3+6x^2+12-1=0. First graph this equation to estimate. Use the estimate for Newton-Raphson

“A real number is rational if and only if it has a periodic decimal expansion”

Define the present usage of the word periodic and prove the statement

1) Determine whether these statements are true or false. Please explain why so I can understand where the answers came from

a) ∅ ∈ {∅}

b) ∅ ∈ {∅, {∅}}

c) {∅} ∈ {∅}

d) {∅} ∈ {{∅}}

e) {∅} ⊂ {∅, {∅}}

f ) {{∅}} ⊂ {∅, {∅}}

g) {{∅}} ⊂ {{∅}, {∅}}

2) Let A = {a, b, c}, B = {x, y}, and C = {0, 1}. Find

a) A x B x C

d) B x B x B

3) Show that if A and B are sets, then

b) (A ⊕ B) ⊕ B = A. (prove every step! you may use the fact that symmetric difference of sets is associative)

Examine the propagation of roundoff through division.

Determine whether the statements in (a) and (b) are logically equivalent.

1. Bob is both a math and computer science major and Ann is a math major, but Ann is not both a math and computer science major.

2. It is not the case that both Bob and Ann are both math and computer science majors, but it is the case that Ann is a math major and Bob is both a math and computer science major.

In a (2, 5) Shamir secret sharing scheme with modulus 19, two of the shares are (0,11) and (1,9). Another share is (3,k), but the value of k is unreadable. Find the correct value of k.

|    | a1   | a2   |
|----|------|------|
| b1 | 0.37 | 0.16 |
| b2 | 0.23 | ?    |

5. Calculate the conditional probability distribution, ?(?|?)P(A|B).

6. Calculate the conditional probability distribution, ?(?|?)P(B|A).

7. Does ?(?|?)=?(?|?)P(A|B)=P(B|A)? What do we call the belief that these are always equal?

8. Does ?(?)=?(?|?)P(A)=P(A|B)? What does that mean about the independence of ? and B?

Using traditional methods it takes 8.2 hours to receive a basic flying license. A new license training method using Computer Aided Instruction (CAI) has been proposed. A researcher used the technique on 26 students and observed that they had a mean of 8.0 hours with a variance of 2.89 . Is there evidence at the 0.1 level that the technique reduces the training time? Assume the population distribution is approximately normal. Step 1 of 5: State the null and alternative hypotheses.

Discribe in 300. words systems like RSA; authentication in public-key systems uses digital signature it was invented in 1977 by Ron Rivest, Adi Shamir, and Leonard. please type

Let G = Z4XZ3XZ2 and consider the two cyclic subgroups H = h(0; 1; 1)i and K = h(2; 1; 1)i of G. (a) Find all cosets (along with the elements they contain) to H and K, respectively. (b) Write down Cayley tables for the factor groups G=H and G=K, and classify them according to the Fundamental Theorem of Finite Abelian Groups.

COMP1805AB (Fall 2019)  "Discrete Structures I" Specification for Assignment 1 of 4

Please ensure that you include your name and student number on your submission. Your submission must be created using Microsoft Word, Google Docs, or LaTeX.

1. Translate the following English expressions into logical statements. You must explicitly state what the atomic propositions are (e.g., "Let p be proposition ...") and then show their logical relation.

   1. If it is red then it is not blue and it is not green.

   2. It is white but it is also red and green and blue.

   3. It is black if and only if it is not red, green, or blue.

2. Determine if the following expressions are tautologies, contradictions, or contingencies by using only truth tables. Show all your work and do not skip any steps (i.e., ensure you that you include a new column for every single operation and that you state at the end whether each expression is a tautology, contingency, or contradiction).

   a. ¬(?∧(¬?∨(?→?)))
   b. (? ∨(? ↔(¬?∧ (? ∨?))))

   c. (? ∧¬(? ∧(? ∨(? ↔?))))d. ¬(?∨¬(?→(?∧?)))

3. Determine if the following expressions are tautologies, contradictions, or contingencies by using only the logical equivalences. Show all your work and do not skip any steps (i.e., ensure you that you include the name of each equivalence used (excluding commutativity and associativity) and that you state at the end whether each expression is a tautology, contingency, or contradiction).

   a. (? ∧(? →¬?))b. (¬?→(?∨?))c. (? ↔(?∨ ¬?))d. (¬? ∨(?∧?))

4. Using only the  and the  operators, find a logical expression that is equivalent to (? → (? ∧ p)) → (? ∨ p). For this question, you do not need to specify "how" you found the equivalent expression because you will show both techniques in questions 5 and 6 below.

5. Prove that the expression you found for question 4 above is equivalent to the expression (? → (? ∧ p)) → (? ∨ p) by using only truth tables. Show all your work and do not skip any steps (i.e., ensure that you include a new column for every single operation).

6. Prove that the expression you found for question 4 above is equivalent to the expression (? → (? ∧ p)) → (? ∨ p) by using only the logical equivalences. Show all your work and do not skip any steps.

7. Let L(x) be the predicate "x is a lion", G(x) be the predicate "x is a giraffe", and M(x) be the predicate "x eats meat". Translate the following expressions into English. The universe of discourse is all animals.

   1. ∀? (?(?) ∧ ?(?))

   2. ∃? (¬(?(?) ∧ ?(?)) ∨ ¬?(?))

   3. ∀? ((?(?) ∨ ?(?)) → (¬?(?) → ?(?)))

(Investment): An investor has $150,000 to invest in oil stock, steel stock, and government bonds. The bonds are guaranteed to yield 5%, but the yield for each stock can vary. To protect against major losses, the investor decides that the amount invested in oil stock should not exceed $50,000. The total amount invested in stock CANNOT exceed the amount invested in bonds by more than $25,000.

a) Set up the problem (decision variables, problem constraints, non-negativity constraints).

b) Now form the objective function if oil stock yields 12% and the steel stock yields 9%. How much should be invested in each alternative in order to maximize the return (don't forget the bonds). What is the maximum return?

Determine if the vector(s), polynomial(s), matrices are linearly independent in R^3, P3(R), M 2x2 (R). Show algebraically how you found your answer.

a. < 2, 1, 5 > , < -2, 3, 1 > , < -4, 4, 1 >

b. x^3 - 3x^2 + 2x +1, -2x^3 + 9x^2 -3, x^3 + 6x

c. | 1 2 | | -3, -1 |

| -4  2 | , | 2 1|

(a) Find the limit of {(1/(n^(3/2)))-(3/n)+2} and use an epsilon, N argument to show that this is indeed the correct limit.

(b) Use an epsilon, N argument to show that {1/(n^(1/2))} converges to 0.

(c) Let k be a positive integer. Use an epsilon, N argument to show that {a/(n^(1/k))} converges to 0.

(d) Show that if {Xn} converges to x, then the sequence {Xn^3} converges to x^3. This has to be an epsilon, N argument [Hint: Use the difference of powers formula].