If I is an ideal of the ring R, show how to make the quotient ring R/I into a left R-module, and also show how to make R/I into a right R-module.

user A sends two messages to user B using ElGamal. user C listens on the line and intercepts both messages, and got the decryption of the first message. explain well if user C can decrypt the second message as well:

1)user A sends two messages E(m) and E(m'). to encrypt the first message user A used the variable k and to encrypt the second k+5. User C knows that there's a difference of 5 between the k values chosen. can he recover the second message?

2)user A sends two messages E(m) and E(m'). to encrypt the first message user A used the variable k and to encrypt the second 2k. User C knows that the second k used is two times the first k used. can he recover the second message?

please explain mathematically and elaborate so i can understand and learn from your answer. thank you very much.

Two sets are separated if the intersection of the closure of one of the sets with the other is empty.

a) Prove that two closed and disjoint sets of some metric space are separate.

b) Prove that two open and disjoint sets of some metric space are separate.

Prove that there are real non-algebraic numbers. Give two examples, the two most famous examples.

2. Some fourth-grade students are practicing reading and comparing decimal numbers. They created two different numbers using decimal squares. Stephanie reads the decimal squares below as “zero and forty-five hundredths” and “zero and one tenth.” She says: “Zero and forty-five hundredths is thirty five hundredths greater than one tenth.” Her partner, Ingrid says: “That’s not right. The square on the left is thirty-five squares bigger than the one on the right.”

a) What does Stephanie seem to understand? What does Ingrid seem to understand?

b) What could you talk about with these students to help improve their understanding?

solve for x:

[x * sqrt(1+x²)] + ln[x + sqrt(1+x²)] = 25

Determine if the statement below is True or False. Justify your answer by giving a proof or counterexample. Let A,B,C∈Mn×n(R) . Suppose C is invertible and C=AB. Then the columns of A, B and C are each bases for Rn and B is the change of basis matrix from the columns of C to the columns of A.

Using Kurosch's subgroup theorem for free proucts,prove that every finite subgroup of the free product of finite groups is isomorphic to a subgroup of some free factor.

HbR is reported to have P50=14torr and a Hillcoefficient=1.2. Calculate ΔYO2 for a climber with HbR assuming that, at 14,000 ft (∼4300m), PO2=50mmHg in lungs and PO2=10mmHg in muscle capillaries.

1.) Suppose that the statement form ((p ∧ ∼ q)∨(p ∧ ∼ r))∧(∼ p ∨ ∼ s) is true. What can you conclude about the truth values of the variables p, q, r and s? Explain your reasoning

2.Use the Laws of Logical Equivalence (provided in class and in the textbook page 35 of edition 4 and page 49 of edition 5) to show that: ((∼ (p ∨ ∼ q) ∨ (∼ p ∧ ∼ r)) ∧ s) ≡ ((r → q) ∧ ∼ (s → p)) where p, q, r and s are statements

Solve the cauchy problem (x^2)*y''-9*x*y'+21*y=0, y'(1)=5 on the interval (1,3). Plot the graphs of y(x) and y'(x)

Please Provide only MATLAB Code.

The Cauchy-Schwarz Inequality Let u and v be vectors in R 2 .

We wish to prove that ->    (u · v)^ 2 ≤ |u|^ 2 |v|^2 .

This inequality is called the Cauchy-Schwarz inequality and is one of the most important inequalities in linear algebra.

One way to do this to use the angle relation of the dot product (do it!). Another way is a bit longer, but can be considered an application of optimization. First, assume that the two vectors are unit in size and consider the constrained optimization problem:

Maximize u · v

Subject to |u| = 1 |v| = 1.

Note that |u| = 1 is equivalent to |u| 2 = u · u = 1.

(a) Let u = a b and v = c d . Rewrite the above maximization problem in terms of a, b, c, d.

(b) Use Lagrange multipliers to show that u · v is maximized provided u = v.

(c) Explain why the maximum value of u · v must, therefore, be 1.

(d) Find the minimum value of u · v and explain why for any unit vectors u and v we must have |u · v| ≤ 1.

(e) Let u and v be any vectors in R 2 (not necessarily unit). Apply your conclusion above to the vectors: u |u| and v |v| to show that (u · v) ^2 ≤ |u|^ 2 |v|^ 2 .

Calculate the final dates on which cash discounts for the following invoices may be taken, and the amount to remit. Invoice total is $5,500 and invoice date is 6/13. Merchandise is received on 6/19. (5 pts.)

a. 6/10 EOM

b. 4/15 n/30

c. 3/10 ROG

Proposition 6.18 (Division Algorithm for Polynomials). Let n(x) be a polynomial that is not zero. For every polynomial m(x), there exist polynomials q(x) and r(x) such that

m(x) = q(x)n(x) +r(x)

and either r(x) is zero or the degree of r(x) is smaller than the degree of n(x).

Prove

Question 2 Materials Requirements Planning An MRP exercise is being implemented over an 8-week period and the following relevant information is provided: One (1) unit of A is made of two (2) units of B, two (2) unit of C, and two (2) units of D. B is composed of two (2) units of E and one (1) unit of D. C is made of two (2) units of B and three (3) units of E. E is made of two (2) units of F. Items B, C, and E have one (1) week lead times; A and F have lead times of two (2) weeks; D has lead time of three( 3) weeks. Assume that lot-for-lot (L4L) lot sizing is used for Items A, B and F; lots of size fifty (50), fifty (50) and one hundred (100) are used for Items C, D and E respectively. Items C, E and F have on-hand (beginning) inventories of twenty (20),fifty (50) and fifty(50), respectively; all other items have zero beginning inventory. We are scheduled to receive ten (10) units of A in Week two (2), fifty (50) units of E in Week one (1), and also one hundred (100) units of F in Week one ( 1). There are no other scheduled receipts.

a) Draw the product structure tree with low level coding [5 marks]

b) Draw the corresponding time-phased diagram showing lead times to scale. [5 marks

] c) If forty (40) units of A are required in Week eight (8), determine the necessary planned order releases for all components {6 schedules}