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Discrete Structures

Discrete Structures refers to the study mathematical structures that are individually separate and distinct rather than continuous. As opposed to the study of calculus or real numbers which deal with continuous variables, Discrete Structures deals with graphs and statements in logic which can be enumerated through the implementation of integers.

Since the study of integers percolate through almost every discipline of Mathematics, Discrete Structures can be more appropriately defined by what it excludes: continuous variables. Thus, in this light, a variety of mathematical topics can be categorized under Discrete Structures, which range from Set Theory to Algebraic Structures such as Rings and Fields to Graph Theory.

Although the main definition and objective is to study discrete objects, modern analytical methods have used Discrete Math to study continuous variables. Thus, understanding Discrete Structures is crucial for not only the study of Discrete Math, but also Mathematics in general, due to both the extensive and expansive dispersion of these objects in other mathematical disciplines.

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Categories within Discrete Structures

Discrete logarithm problem

Consider the multiplicative group of integers 1, 2,...,10, modulo 11^5. We want to compute the power we need to raise 2 to get 3.

characteristic functions of subsets satisfy the properties

Prove that the characteristic functions of subsets satisfy the foowing properties: (a) f_(A intersection B) (x) = f_A (x) f_B (x) for all x, (b) f_(A union B) (x) = f_A (x) + f_B (x) - f_A (x) f_B (x) for all x, (c) f_(A symmetric difference B) (x) = f_A (x) + f_B (x) - 2f_A (x) f_B

Solving discrete math problems

1) Use Venn diagrams to determine whether each of the following is true or false: a. (A union B) intersect C = A union (B intersect C) b. A intersect (B union C) = (A intersect B) union (A intersect C) 2) Calculate the number of integers divisible by 4 between 50 and 500, inclusive. 3) Use the permutation formula to

Steps on Solving Discrete Questions

I need some help with this mathematical questions: (a) Use the Binomial Theorem to write the expansion of (x + y) 6? (b) Write the coefficient of the term x2y4z5 expansion of (x + y + z) 11. Let A = {a, b, c, d}, and let R be the relation defined on A by the following matrix: (see attachment for (a) Describe R by list

Discrete Math - Structures

Make a meaningful example illustrating Project Evaluation and Review Technique (PERT). Explain all details. Draw the graphs involved. Use reasonable number of tasks. (a) Determine the minimum time needed to complete your set of tasks. (b) Determine the Critical Path for your example. Explain what this gives you. (c) Comment

The solution gives detailed steps on solving a sets of finite math problems. All formula and calculations are shown and explained if necessary.

Write the following set in builder notation form. { 7, 12, 17, 22, 27, 32, 37, 42, 47, 52 }

Binomial coefficients and nonnegative integers

Prove the identity (n~r) (r~k) = (n~k) ((n-k)~(r-k)), whenever n, r, and k are nonnegative integers with r less than/equal to n and k less than/equal to r: a) using a combinatorial argument. b) using an argument based on the formula for the number of r-combinations of a set with n elements.

Digraphs, Closure of Relations, and Hasse Diagrams

Please provide some guidance on this exercise. The more I read about digraphs, closure of relations, and Hasse diagrams the more confused I get. It is just not sinking in. Please help and explain this text exercise.

Frequency Table, Relative Frequency and Cumulative Frequency

Use 4 point bins 96 to 99, 92 to 95 etc. to make a frequency table for the set of exam scores. Include columns for relative frequency and cumulative frequency. Scores: 97, 80, 90, 95, 78, 98, 89, 79, 99, 86, 98, 78, 93, 90, 78, 98, 79, 77, 98, 89

String Bits

How many bit strings of length n are palindromes? Hint: Consider two cases n is even and n is odd. Note a palindrome is a "string" of letters or numbers which read the same frontwards and backwards. Examples: MOM, 1101011, 10111101 are palindromes.

Big-O, Big-Theta

1. Using the definition of big-O, show that 2n element of O(6*n^2) 2. Using the definition of big-O, show that 6n^2 not element of O(2n) 3. Show that 6n^2 element of THETA(n^2) Do not use limits, just the big O and big Theta definitions. Please show the definitions.

Frequency for Manage Position Control Classification

1.3. MANAGES POSITION CONTROL (Classification) * The reported monthly frequency for your location is 63, which equates to 147.2 times per month per 1000 civilians serviced by the CPS. * The average frequency for locations within two standard deviations from the mean is 28 times per month for every 1000 civilians service

Discrete Structure

(a) Define the function f: R-->R by f(x) = x3 + 4. Briefly explain why f is a 1-1 (one-to-one) function. No proof necessary, just an explanation in some detail (b) Is the function g: R -->Z defined by g(n) = [n/2]a one to one function? (Be careful,[n/2]means the ceiling function.) Explain. (c) Briefly explain what f-1 means in

Hasse Diagram - Ordered Pairs

Please see the attached. Consider the following Hasse diagram of a partial ordering relation R on a set A: (a) List the ordered pairs that belong to the relation. Keep in mind that a Hasse diagram is a graph of a partial ordering relation so it satisfies the three properties listed in number 5 part(b). (b) Find the (Boo

standard scores

Consider a normal population with µ = 25 and Ï? = 8.0. (A) Calculate the standard score for a value x of 23. (B) Calculate the standard score for a randomly selected sample of 45 with X(line on top)= 23. (C) Explain why the standard scores of 23 are different between A and B above.

Set up initial Simplex Tableau and determine the first pivot.

Minimize w = 29y1 + 10y2 subject to: 3y1 + 2y2 is greater then or equal to 2 5y1 + y2 is greater then or equal to 3 with y1 is greater then or equal to 0, y2 is greater then or equal to 0 Set up the inital Simplex Tableau and determine the first pivot. Please show all s

Recursion for Recursive Functions

Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of non negative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a non negative integer and prove that your formula is valid. a) f(0) = 1,f(n) = - f(n - 1) for n >= 1 b) f(0

Recursion Functions

4. Find f(2), f(3), f(4), and f(5) if f is defined recursively by f(0) = f(1) = 1 and for n = 1,2, ... a) f(n + 1) = f(n) - f(n - 1). b) f(n + 1) = f(n)f(n - I). c) f(n + 1) = f(n)^2 + f(n - 1)^3. d) f(n + 1) = f(n)/f(n - 1).

Pigeonhole Principle Examination

You pick cards one at a time without replacement from an ordinary deck of 52 playing cards. What is the minimum number of cards you must pick in order to guarantee that you get (a) a pair (example, two kings or two 5s), (b) three of a kind (example, three 7s)

Countable or uncountable numbers

Given a set of all real numbers containing 4s and 5s in their decimal representation. Determine the given set is countable or uncountable. Example : numbers in this set: 4.4455544, 455.55554444 You don't need to prove this, but rather explain it to me why. I understand why from 0-1 is uncountable or from 1-2 etc using

Truth Statements and Conditionals

18. Let p represent a statement having a truth value T, q represent a statement having a truth value F, and r represent a statement having a truth value T. Find the truth value of each of the following... 4. Determine the truth of the following conditionals... Construct the truth tables for the statements expressed in Exercis

Inverses of modulus

A: Show that the positive integers less than 11, except 1 and 10, can be split into pairs of integers such that each pair consists of integers that are inverses of each other modulo 11. Use part (a) to show that 10! is congruent to -1 (mod 11).

The Minimum of Two Independent Geometric Random Variables

Let H be a random variable with probability p1 that has a Geometric distribution G1 Let Y be a random variable with probability p2 that has a Geometric distribution G2 H and Y are two independent geometric random variables. What is distribution of Z =min(H,Y)

Determine if Simplex Tableau is in Final Form

If it is in final form, what is the programming problem. If it is not is final form what is the pivot element? x y u v p constant 1 1 1 0 0 6 1 0 -1 1 0 2 ---------------------- 3 0 5 0 1 30

How many bit strings of length 6?

(a) How many bit strings of length 6 are there? Explain. (b) How many bit strings of length 6 are there which begin with a 0 and end with a 0? Explain. (c) How many bit strings of length 6 start with a 0 bit or end with a 0 bit? Explain. (d) How many bit strings of length 6 are there which contain exactly 3 ones? Exp

Arbitrary Number Conjecture

1. Pick any number and add 10 to the number. Divide the sum by 5. Multiply the quotient by 5. Subtract 10 from the product . Then subtract your original number. a. What is the result? b. Arbitrarily select some different numbers and repeat the process. c. Can a conjecture be made regarding the result when this process is fo

Manufactures

Raul Mondesi manufactures and sells homemade wine, and he wants to develop a standard cost per gallon. The following are required for production of a are required for production of a 50 gallon batch. 3,000 ounces of grape concentrate at \$0.04 per ounce. 54 pounds of granulated sugar at \$0.30 per

Root mean square deviation question

If there are 50 pieces of data which range from 0 to 100: - with a mean of 60 and Root mean square deviation of 8 show that only 1 piece of data can equal 100 -with a mean of 60 and Root mean square deviation of 8.2 show that only 2 (or less) pieces of data can equal 100 -with an Root mean square deviation 4&#8730;2 or less s

Discrete Structures Math Problem?

Let A = { 1, 2, 3, 4 } , and let R be the relation defined on A defined by: R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,3), (3,4), (1,3), (2,4)} (a) Draw the digraph of this relation. (b) Which of the properties: reflexive, antisymmetric and transitive are true for the given relation? Begin your discussion by defining each

magnitude of forces

A rectangular gate ABCD of mass 120kg where AB=1m and AD=3m is supported at A & B by pins where B is vertically above A. The reaction force at B is always horizontal. A person with mass 45k sits on the gate at C. Find the magnitude of forces A & B using moments.