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Discrete Math

Symmetric Generalized Eigenproblem: Orthogonality Proof

For the s y m m e t r i c g e n e r a l i z e d e i g e n p r o b l e m A y = B y , l e t ' s p r o v e t h e o r t h o g o n a l i t y s t a t e m e n t s y JT B y i = 0 , J≠ i a n d yJT A y i = 0 , J ≠ i f o r J = i . F i r s t , w r i

Discrete Math : Functionally Complete

6. a) What does it mean for a set of operators to be functionally complete? b) Is the set {+, .} functionally complete? c) Are there sets of single operator that are functionally complete? Please see the attached file for the fully formatted problems.

Fixed Point Iteration Problem Using Matlab

Given the function f(x) =1 + 0.5 sin x, show that it has a unique fixed point x* in the interval I=[0,2] and that the iterations xn+1=f(xn) converge to the fixed point for any x0 &#1028; I. Also, find the number of iterations necessary to guarantee that |xn - x* | < 10^-2 . Write a short Matlab code to find the fixed po

Discrete Math : Counting and Relations

Basics of Counting 32. How many functions are there from the set {1, 2, ... , n}, where n is a positive integer, to the set {0, 1}? Relations & Relation Properties 24. Let R be the relation R = {(a, b) | a < b} on the set of integers. Find a) R -1 b) bar-R Application of Relations 8. Suppose that R is a symmetric

Rules of Inference, Logic and Symbology

Designate each simple statement with a letter. Then write down the compound statements using the following rules(modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, addition, simplification, or resolution), arrive at the conclusion. I've done all but the latter. If you send an email then I will write t

Domain and image sets

If I have the function f(x)=2e x+1 , how would I discover the domain and image set of function f? How would I go about finding the domain and image set of f -1? How would I go about solving the equation y= 2e x+1, to find x in terms of y?

Homotopy

(See attached file for full problem description with proper symbols) --- Let X and Y be topological spaces and let be a subspace. Show that 'homotopic relative A' defines an equivalence relation on the set of continuous maps from X to Y which agree with some fixed map on A. ---

Discrete math proofs..

Proper walk through of following proofs required ( for a better understanding ) --- 1) Prove that if n is an odd integer then n2 = 1 mod 8 2) Prove that 5n+3 is divisible by 4 for all integers n>=0

Discrete math proofs

(See attached file for full problem description with proper symbols and equations) --- 1)Prove that for any non-empty sets A x (B-C) = (AxB)-(AxC) 2) Let a,b be integers and m a positive integer. Prove that: ab = [(a mod m ) * (b mod m) mod m ] 3)Prove or disprove (a mod m) + (b mod m) = (a+b) mod m for all intege

Uniform Continuity : Epsilon-delta Proof of Continuity

Prove (or disprove) the following statement: A function f exists that is uniformly continuous on (a,&#8734;) and for which lim as x-> &#8734; of f(x) = &#8734;. I know that f(x) = x ^ (1/3) (cube root of x) is uniformly continuous on R, and that it's limit as x approaches infinity is infinite. However, I am having troubl

Discrete Mathematics and its Applications : Greedy Algorithms

Greedy Algorithms procedure change (c1, c2, ...., cr: values of denominations of coins, where c1 > c2 > ... > cr; n: a positive integer) for i : = 1 to r while n &#8805; ci begin add a coin with value ci to the change n :=n - ci end 2. Use the greedy algorithm to make change using quarters, dimes, ni

Discrete Mathematics and its Applications : Greatest Common Divisors

A) Define the greatest common divisor of two integers. b) Describe at least three different ways to find the greatest common divisor of two integers. c) Find the greatest common divisor of 1,234,567 and 7,654,321. d) Find the greatest common divisor of 2335577911 and 2937557313. Please see the attached file for the fully f

Matlab Function for Cubic Polynomial Interpolation Algorithm

(See attached file for full problem description) --- There are a few parts to this problem. Please complete and explain each part. 1. Consider the array in which, for some fixed x, the , , and are computed by the formulas Write a Matlab function which will take inputs , and will

Discrete Mathematics and its Applications - Sets and logic

For each topic demonstrate a knowledge and capability by giving the following information: 1) Problem Solution: (solution for an even number problem) See below... 2) Personal Observation: (personal comment on the topic including advice to others on how to study and understand it). Logic 10. Let p, q, and r be the prop

Set Theory : Pairwise Disjoint Finite Sets and Addition Principle

Proposition 10.2.1: (the addition principle) Suppose that X and Y are disjoint finite sets. Then X U Y is finite and |X UY| = |X| + |Y|. Corollary 10.2.2: For a positive integer n, suppose that X1, X2....,Xn is a collection of n pairwise disjoint finite sets (i.e. i does not = j => Xi Xj = empty set) Then X1 U X2

Set Theory Proof : Inclusion-Exclusion Principle

3. This exercise is about the inclusion-exclusion principle. a) Let X and Y be finite ts and suppose that |X| = 11, |Y| = 6, and |X&#8745;Y| =4. Find |XUY|. b) Suppose that U is a finite universal set. If |U| = 21, |XUY| = 11. |X| = 4 and |Y|= 10. find |XcUYc|. c) Each tile in a collection of 19 is a square or a triangle and

Dynamic Programming : Write an Algorithm to Minimize Cost

There are n trading posts along a river. At any of the posts you can rent a canoe to be returned at any other post downstream. (It is next to impossible to paddle against the current.) For each possible departure point i and each possible arrival point j the cost of a rental from i to j is known. However, it can happen that

Logic Example Problem

This problem is about the proof of Theorem 1 implies Theorem 2 as discussed in class. Regard Theorem 1 as a statement P and Theorem 2 as the statement "Q implies R". Then the statement "Theorem 1 implies Theorem 2" can be expressed as: "P implies (Q implies R)". Theorem 2" is can be expressed as P implies (Q implies H)". a

Algorithm

Let T[1..n] be a sorted array of distinct integers, some of which may be negative. Give an algorithm that can find an index i such that 1 <= i <= n and T[i] = i, provided such an index exists. Your algorithm should take a time in Big "O" (log n) in worst case.

Correspondence of Borel sets

If f is one-to-one, f, f^-1 are continuous, then f is called a homeomorphism. Now I want you to prove the following: Let f : X -> Y, ( X and Y are topological spaces)be homeomorphism, prove that it establishes one-to-one correspondence between Borel sets in X and Y.

Proof of Differentiability

Let f(x) = { x^2 if x is rational { 0 if x is irrational Show that f is differentiable at x=0 but not at any other point. --- Please see the attached file for the fully formatted problem.

5 Finite Math Problems

1) A man walking in the woods encounters a stream. Because he is unsure of the stream depth, he measures how deep the water is in many random spots along the entire width of the stream. After 1000 measurements each with a depth of 6 inches, he concludes the probability of the water being 6 inches deep the entire way across to

Finite Math

1.) Dan borrows $500 at 9% per annum simple interest for 1 year and 2 months. What is the interest charged, and what is the amount due? ________________________________________ 2.) A mutual fund pays 9% per annum compounded monthly. How much should I invest now so that 2 years from now I will have $100 in the account?