Prove That For All Natural Numbers A > 1 And B > 1, If A Divides B Then A Does Not Divide B+1 (prove by contradiction)

Consider the following linear program. Maximize z= 5x1+ 3x2

subject to 3x1+ 5x2≤15

5x1+ 2x2≤10

– x1+ x2≤2

x2≤2.5

x1≥0, x2≥0

a. Show the equality form of the model.

b. Sketch the graph of the feasible region and identify the extreme point solutions. From this representation find the optimal solution.

c. Analytically determine all solutions that derive from the intersection of two constraints or nonnegativity restrictions. Identify whether or not these solutions are feasible, and indicate the corresponding objective function values. Which one is optimal?

d.Let the slack variables for the first two constraints, x3and x4, be the axes of the graph, and sketch the geometric representation of the model. Show an iso-objective line in these variables, and from it determine the optimal solution.

Topology question:

Prove that a bijection f : X → Y is a homeomorphism if and only if f and f−1 map closed sets to closed sets.

Use induction to prove that 8^n - 3^n is divisible by 5 for all integers n>=1.

1. The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide Vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 48 milligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes 6 milligrams of A and 2 milligrams of B. An ounce of oats costs $0.05, and an ounce of rice costs $0.03.

1. Formulate a linear programming model for this problem.
2. Solve the model using graphical analysis.
3. What would be the effect on the optimal solution in the above problem if the cost of rice increased from $0.03 to $0.06 per ounce?

The size of NMOS is 4lambda/2Lamda. Size PMOS for

a) Symmetric performance

b) Optimum Performance

And then calculate the propogation delay for a chain of four inverters. Assume a 0.25 microns technology.

(a) Briefly describe what is meant by the word cryptography.

(b) Briefly describe the Vigenere cipher, including a discussion of the encryption and decryption processes.

(c) Describe what is meant by a ‘Feistel Cipher’.

(d) DES includes S-boxes as part of its encryption and decryption steps. Each of eight S-boxes is a fixed 4 × 16 array, whose entries come from the integers 0, 1, . . . , 15. Describe in detail how DES transforms a 48-bit string into a 32-bit string using the S-boxes.

(e) Without giving details of the algorithm, briefly discuss some of the shortcomings of DES.

THIS QUESTIONS FOR CRYPTOGRAPHY

this is Discrete mathematics problem.

Translate the following English sentences into propositional formulas. Remember to
def ne your atomic propositions!
(a) Either the suspect wore gloves, or he didn't touch the doorknob.
(b) I will eat my tie if the Cubs win the World Series.
(c) It smelled funny, but he ate it anyway.
(d) The people will give up their arms only when the tyrant resigns and we get our
money back.
(e) All prizes will be awarded provided that a su cient number of eligible entries are
received.
(f) The absence of antibodies in the subject's body implies that they are not susceptible
to infection.
(g) Participants were not timed on this task; however, most finished in less than 8
minutes.
(h) The printer can hold a black ink cartridge or a color cartridge, but not both.
(i) Only if a person has a real desire to want to change and puts forth real e ort to
make those changes do they come about.
(j) The existence of a Lyapunov function is a necessary and su cient condition for the
stability of the system.

(k)The new regulation does not apply unless both houses vote to impose it.

I got a wrong answer for many of case, and English is not my first language, so it is little hard to understand concept.

please help me.

• Thread 1: Why do you think that women are still continued to be viewed as objects in most, if not all, media? Has this situation changed much from the past to present day? How does this pertain to the male gaze today?  

• Thread 2: By all indications, including the 2016 election, women continue to be sidelined, stereotyped and sexualized in popular media and entertainment. Girls and boys from a young age seem to have been given a certain outlook on how women should act, dress and the type of jobs that they can obtain, all through media. With this still being a major issue, what would you suggest is the best or most efficient way to have women not be sidelined--in media, politics, education, or any other aspect of life--that we can instill in all children at a young age and create a new “norm” for our youth? Can we instill a new "norm"?

prove that cube root of 26 is irational

Question 1. Let F be an ordered field. For each of the following statements, prove the statement or provide a counterexample.
(a) For all x,y,z,w ∈F, if x < y and xw < yz, then w < z.

(b) If x,y,z,w ∈F, then |x + w|≤|x + y|+|y + z|+|z + w|

Let x ∈R, a ∈R, and b ∈R.
(a) Suppose that |x−a| = 3|x−b|. Let

c =(9b−a)/ 8
. Prove that |x−c| = 3 8|a−b|

Let G = Z x Z and H = {(a, b) in Z x Z | 8 divides a+b}

a. Prove directly that H is a normal subgroup in G (use the fact that closed under composition and inverses)

b. Prove that G/H is isomorphic to Z8.

c. What is the index of [G : H]?

1. You are a salesperson for Bottoms Up Beverage Company and are introducing a new bottled water product called Fresh-is-Best Sweetwater, sourced from the natural springs of the Great Lakes near Flint, Michigan. You are preparing to call on the GO GO! Convenience store chain in an attempt to gain authorization for Fresh-is-Best Sweetwater distribution and floor displays in all 85 of the chain’s outlets. Your regular wholesale price (to the retailer) is $16.80 per case of twenty-four 20 oz. bottles. a. In order for GO GO! to obtain its standard margin of 44%, it would need to establish a price to consumer of ____________per 20 oz. bottle.

The chain estimates that it will sell 10 cases per store (240 bottles) each week at this price. However, to generate strong consumer trial of your new product, you would like to convince the chain to authorize large Fresh-is-Best Sweetwater displays in each store and offer an introductory price to consumer of 99 cents. To facilitate this, Bottoms Up Beverage is offering a “Get Rolling” promotional $12.00 per case cost to the retailer when the store buys 50 cases or more at a time and fulfills certain merchandising requirements (e.g., displays and advertising). b. How many cases per week would the chain have to sell in each store to break even and make this proposition viable for GO GO!? ____________

Although you hope to sell GO GO! on this program, Bottoms Up also offers a slightly less attractive deal of $14.40/case (“Get Started” promotion) with a 20-case minimum per store purchase and less aggressive merchandising requirements. c. How many cases would each store have to sell to break even in this scenario if it still ran a 99 cent introductory price to consumer? _________________. What about if it ran a 1.09 sale price instead? __________________

In test markets, Bottoms Up found that retailers that took advantage of Bottoms Up’s $12.00/case “Get Rolling” deal and merchandised Fresh-is-Best Sweetwater 20 oz bottles as per the requirements with a 99 cent price to consumer generated on average a 75% increase in sales over those stores that just priced the product at the regular price to consumer of $1.25. d. Assuming GO GO! experiences similar results, what would the total incremental profit impact be to the GO GO! chain if it chooses to take advantage of the “Get Rolling” promotion and offer a consumer price of 99 cents versus agreeing to the “Get Rolling” promotion but offering a consumer price of $1.25? ___________ What would the total profit impact be to the GO GO! chain if it chooses to not take advantage of the “Get Rolling” promotion and instead buy Fresh-is-Best Sweetwater at the regular price of $16.80/case and sell it to consumers for $1.25/bottle? ____________

A composition of n is made by breaking n down into summands. For example, the compositions of 3 are {3}. {2 + 1}, {1 + 2}, {1 + 1 + 1}. In general, there are 2^(n-1) compositions of n. Prove that there are 3^(n-1) double compositions of n.

Prove A (bar on top of A)= collection of all adherence pts of A where A <=X and (X,d) is a metric space