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Discrete Math

Analytic Zeros Proof

Let f be analytic on a domain D. Prove that if f(z(0))=0 and if f is not identically zero, then z(0) is a zero of f of some finite order m.

Propositional Logic

For the following expression, give one interpretation that makes it true and one interpretation that makes it false: {see attachment for expression}

Propositional Logic : Blood Types

Let the propositions have the intuitive meanings given in the Figure attached. Write a clause or product of clauses that express the following ideas. (a) If test T is positive, then that person has blood type A or AB (b) If test S is positive, then that person has blood type B or AB (c) If a person has type A, the test

Discrete f3

If there is any reason why you do not want to answer the question (problem with attachment, bid price, ect.) please let me know.

Discrete Math : Proof that there must be 12 Pentagons on a Soccer Ball

A soccer ball is formed by stitching together pieces of material that are regular pentagons and regular hexagons. Each corner of a polygon is the meeting place for exactly three polygons. Prove that there must be exactly 12 pentagons. (Please see attachment for full question and background)

Propositional Logic : DeMorgan's Laws and Truth Tables

Please see the attached file for the fully formatted problems. Verify DeMorgan's laws (equation 1 and 2 below) using truth tables. Prove the generalized DeMorgan's laws: (1) (NOT(p1 p2 .... pk)) (2) (NOT(p1+p2+...+pk)) by induction on k, using the basic laws: NOT(pq) NOT(p+q) Then, justify the ge

Newton's Method Proof

Please show that when n=1, Newtons method given by: x^k=x^(k-1)-(J(x^(k-1))^-1)(F(x^(k-1)) for k>=1 reduces to the familiar Newton's method given by: P_n=P_n-1 - f(p_n-1)/f'(P_n-1) for n>=1 Note: ^-1 is inverse J is the jacobian matrix The top equation is called newton's method for non linear systems. x is a vecto

Discrete 47.3

3. Let d1,d2...dn be .... prove that d1...dn are degrees of the vertices. (see attachment for full question)

John Nash's Game Theory

Can anybody explain and summarize the detail of John Nash's paper please? It is in the attachment file. Thank you.

Chinese Remainder Theorem and Proofs

The Chinese Remainder Theorem (CRT) applies when the moduli ni in the system of equations x≡ a1 (mod n1) ... x≡ ar (mod nr) are pairwise relatively prime. When they are not, solutions x may or may not exist. However, the related homogeneous system (2'), in which all ai=0, always has a solution, namely the trivial

Nonnegative Integers

If the solution to this nonnegative integer question is correct, then you may respond that it is. If the solution needs ANY kind of improvement, in presentation, in clarity, in correctness, if a proof can be more elegant, then please rewrite the entire solution.

Analyticity Proof

Suppose that f: C->C and that f is analytic at a point z0 element of C. Prove that there exists a real number r>0 such that, the nth derivative of z0=[n!/(2 pi r^n)]x[int(e^(-niy)f(z0+re^(iy)) from 0 to 2pi with respect to y for all n element of Natural numbers.

Divisors and relative primes

Let a be an integer. Prove that 2a + 1 and a^2+ 1 are relatively prime. ( relative primes are numbers that their largest common divisor is 1).

Word problem

Take an in-depth look into this proof. Obviously it is wrong. Where is it wrong and why? This is obviously wrong. Where and why? Detailed explanation is needed of where and why it is wrong with all examples. Thanks Let a = b. Multiply both sides by a (OK because we don't violate the equal sign). We get a² = ab. Su

Finite Math - Problem with Excel Spreadsheet Attachment

At the start of the year, a company wants to invest excess cash in one-month, three-month and six-month Certificates of Deposit (CD's). (Purchase price and yields for the different CD's appear in the table below). The company is somewhat conservative, however, and wants to make sure that it has a safety margin of cash-on-hand e

Finite Math - Probability

I need assistance with following problem along with steps to arrive at the solution/answer. Three envelopes are addressed for 3 secret letters written in invisible ink. A secretary randomly places each of the letters in an envelope and mails them. What is the probability that at least 1 person receives the correct letter?

Finite Math - Sets Counting Techniques

The problem I am working is the following; please provide step-by-step to obtain solution's. I am unable to figure (c) out the ANSWER IS ... 7805 There are 5 rotten plums in a crate of 25 plums, How many samples of 4 of the 25 plums contain at least one rotten plum?

Recursive definition

We can define sorted lists of integers as follows: Basis - A list consisting of a single integer is sorted. Induction - If L is a sorted list in which the last element is a and if b >= a, then L followed by b is a sorted list. Prove that this recursive definition of "sorted list" is equivalent to our original, nonrecurs

Matlab : Topography of a Tank

I would like to use Matlab to make a graph of the topography of the tank. I would like some one to show me how to do it because I don't understand how to plot all the points. Can someone give me a 3D graph of the topography? In the attached file row one is my X-axis in cm and my y-axis is 1-41.

Binary tree

A. Write 3n − (k + 5) in prefix notation: ????. b. If T is a binary tree with 100 vertices, its minimum height is ????. c. Every full binary tree with 50 leaves has ???? vertices.

Divisibility

Suppose A divides N and B divides N. Does this always imply: A * B divides n? Now the question is under what condition A*B will always divide N? Prove it.

Relations on Set {0,1} : Binary, Reflexive and Symmetric

I have done several examples but these I cannot get right, I am not sure where I have made the mistake and I am confusing myself. a. List all the binary relations on the set {0,1}. b. List the reflexive relations on the set {0,1}. c. List the symmetric relations on the set {0,1}.

Big-Oh Proving Question

A. Use the definition of big-oh to prove that 3n - 8 - 4n^3 / 2n - 1 is O(n^2). B. Use the definition of big-oh to prove that 1 . 2 + 2 . 3 + 3 . 4 + ... + (n - 1) . n is O(n3).

Factoring

A. Convert (11101)2 to base 16. b. Use the Euclidean algorithm to find gcd(34,21).

Recursive definition

I need to give a recursive definition with initial condition(s). a.) The sequence {an}, n = 1,2,3,... where an = 2n. b.) The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, ....