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Discrete Math

Discrete Math: Matching Zeros

Two Social Security numbers match zeros if a digit of one number is zero if the corresponding digit of the other is also zero. In other words, the zeros in the two numbers appear in exactly the same positions. For example, the Social Security numbers 120-90-1109 and 430-20-5402 have matching zeros. Prove: Given a collection

Discrete Math and Statistics: Auto Regressive (AR) Formula

1. Can Var(X_n) = (a^(2n - 2) + A^(2n - 4) + ... + A^2 + 1)sigma^2, be written as[ (sigma^2 )*(SUM(A2)n)from A=0 to A=n-1 =(sigma^2 )*(1/(1- A2 ) if n is large? 2. Why does Cov (Xi,Xj) = 0 if i /= j and sigma^2 if i=j 3. What is the basic equation for the Covariance?

Equivalence Relation vs. Equivalence Class

Concerning discrete math, I am very confused as to the relationship between an equivalence relation and an equivalence class. I would very much appreciate it if someone could explain this relationship and give examples of each such that the relationship (or difference) is clear.

Discrete Math - Counting

Please show these solutions in great detail with all steps explained as they will serve as a guide for future problems. 1. The integers 1 through 25 are arranged in a 5 x 5 array (we use each number from 1 to 25 exactly once). All that matters is which numbers are in each column and how they are arranged in the columns. It do

Discrete Math- Equivalence Relations

Please help with the following problem regarding discrete math. I need a clear explanation of what an equivalence relation is with an examples. Specifically given 5|(m-n), where m and n are integers, please verify if this is an equivalence relation. Please explain this clearly and in detail.

Discrete Math - Relations

Let B = {0,{1},{2},{1,2}}; we define a relation on B. The pairs are of the form (X.Y) with X and Y subsets of {1,2}. We set XRY if absX = absY. Find the relations. Please explain the answer clearly.

Geometric Progression

A geometric progression has first term log_2 27 and common ratio log_2 y, find the set of values of y for which the geometric progression has a sum to infinity. My logic: If sum to infinity exists then 1- log_2 y > 0 - log_2 y > -1 log_2 y < 1 y< 2^1 y< 2 Where next?

Set Theory and Counting

1. List the ordered pairs in the equivalence relations produced by these partitions of {0,1,2,3,4,5} a) {0}, {1,2}, {3,4,5} b) {0,1}, {2,3}, {4,5} c) {0,1,2}, {3,4,5} d) {0}, {1}, {2}, {3}, {4}, {5} 2. Which of these collections of subsets are partitions of the set of integers? a) the set of even integers and the set of

Counting and Set Theory

How many ways can n books be placed on k distinguishable shelves? a) if the books are indistinguishable copies of the same title? b) if no two books are the same, and the positions of the books on the shelves matter? How many ways are there to deal bands of five cards to each of six players from a deck containing 48 differe

Perform Singular Value Decomposition

I need singular value decomposition of the matrix A=[ 1 4 2 -2 -1 3 0 3 5] Split it in 3 matrices. I need a detailed, explicit solution please. Matrices don

Set Theory: Surjective and Bijective

i) If A is countable and f: A=>B is surjective, show that B is countable. ii). Show that a function f: A=>B is bijective if, and only if, there is a function g: B=>A with gf = 1_A and fg = 1_B. iii) If f: A=>B, g: B=>C and h: C=>D are functions show that h(gf) = (hg)f. iv) Let f: A=>B and g: B=>C be functions. a)

Discrete math problems

1. A telephone number is a ten-digit number whose first digit cannot be a 0 or a 1. How many telephone numbers are possible? 2. A computer operating system allows files to be named using any combination of uppercase letters (A-Z) and digits (0-9), but the number of characters in the file name is at most eight (and there has t

Discrete Math: Integers

A US Social Security number is a nine-digit number. The first digit (or digits) may be zero. a) How many US Social Security numbers are available? b) How many US Social Security numbers are odd? c) How many US social security numbers read the same backward and forward (eg 350767053)? These 4-digit numbers cannot start wit

Discrete Math: Truth Tables

1) Construct the Truth Table for each of the following Boolean expressions: Are they equivalent expressions? Are they tautologies? Contradictions? 2) Find a Boolean expression involving x y which produces the following table: 3) Consider a statement of the form "if A then (B and C)". Assume you wish to disprove it. Th

Why Finite Mathematics?

Please explain two things that I should learn before taking a Finite Mathematics class and explain those two things and why they would help me in real life or modern jobs.

Rate the importance of the following issues

In a survey conducted by a union, members were asked to rate the importance of the following issues: (1) job security, (2)increased fringe benefits, and (3) improved working conditions. Five different responses were allowed for each issue. Among completed surveys, how many different responses to this survey were possible? E

Finite Mathematics (Future Value - Present Value)

Can you help me with the following questions: - What is the difference between the accumulated amount (future value) and the present value of an investment? Please give examples of each. - Find the accumulated amount at the end of 8 mo on a $1200 bank deposit paying simple interest at a rate of 7%/year. - A bank depos

Game Theory of Domino

A domino can be placed on a 2x3 board in seven different ways. The first player places the domino and the second player selects one of the six squares. If the selected square is covered by the domino, then the second player wins; otherwise the first player wins. What are the optimal strategies, and what chance do they give

Proof lipschitz continuous

a) Let M>0, and let f:[a,b]-->R be a function which is continuous on [a,b] and differentiable on (a,b), and such that |f'(x)| <= M for all x belonging to (a,b) (derivative of f is bounded). Show that for any x,y belonging to [a,b] we have the inequality |f(x)-f(y)| <= M|x-y|. *apply mean value theorem. b) Let f:R-->R be a di

Logic statements

Determine which, if any of the three statments are equivalent. give a reason for your conclusion. Show complete work. 1) Gasoline costs $1.99 per gallon if and only if you live in Orange Country. 2) You do not live in Orange County and gasoline does not cost $1.99 per gallon. 3) if you do not live in Orange County then gas

logic based statement

Determine which if any of the three statements are equivalent. 1)If the carpet is not clean, then Sheila will run the vacuum. 2)If it is not case that both the carpet is clean and Sheila will run the vacuum. 3)If the carpet is clean, then Sheila will not run the vauuum.

Weights of Codewords of a Binary Linear Code

The weight of w(x) of a vector x in (F_q )^n is defined to be the number of nonzero entries of x. Prove that, in a binary linear code, either all the codewords have even weight or exactly half even weight and half odd weight.

Proof using intermediate value theorem

Let f : [0,1]-->[0,1] be a continuous function. Show that there exists a real number x in [0,1] such that f(x)=x (apply the intermediate value theorem to the function f(x) -x). This point x is known as a fixed point of f, and this result is a basic example of a fixed point theorem.

Standard limit proof

For any x > 0, we have lim n-->infinity of x^1/n = 1 Hint: Treat cases x >= 1 and x < 1 separately. You might wish to first use the following to prove the preliminary result that for every epsilon > 0, and every real number M > 0, there exists an n such that M^1/n <= 1+epsilson: "Let x be a real number. Then the limit lim

Proof of Zero Test for Sequences

Let (a_n) evaluated from n=M to infinity be a sequence of real numbers. Then the limit lim as n-->infinity of a_n exists and is equal to zero if and only if the limit lim as n-->infinity of the absolute value of a_n exists and is equal to zero. Prove and answer if it is still true if we replace zero in the statement above by

Translation of Sentences into Symbolic Predicate Logic

For each of problems, ? Write a symbolic version of the given statement ? Construct a negation of the symbolic statement ? Translate the symbolic negation into good, lucid English. Problem A: "All the routers in our facility support both hard-wired and wireless Internet connections." Problem B: "Each of our salespersons h

Absolute Convergence Test

Prove the absolute convergence test: Let the sum from n=m to infinity of a_n be a formal series of real numbers. If this series is absolutely convergent, then it is also conditionally convergent. Furthermore, in this case we have the triangle inequality - the absolute value of the sum from n=m to infinity of a_n <= the sum fr

Discrete Math

The licensing department issues plates with three letters followed by three digits for cars, and two letters followed by four digits for trucks, except that there are 137 "objectionable" three-letter prefixes that cannot be used. How many vehicles can be licensed?