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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.

Categories within Discrete Structures

Recurrence Relation

Postings: 109

A recurrence relation is an equation that recursively defines a sequence, once one more initial terms are given.

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

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.

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

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

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

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


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

Setting standards: Preparing a Standard Cost Card

Svenska Pharmicia, a Swedish pharmaceutical company, makes an anticoagulant drug. The main ingredient in the drug is a raw material called Alpha SR40. Information concering the purchase and use of Alpha SR40 follows: Purchase of Alpha SR40: The raw material Alpha SR40 is purchased in 2-kilogram containers at a cost of 3,000 K

Combinatories and Tree Diagram

5 coins are tossed a)in how many ways will the first coin turn up heads and the last coin turn up tails b)draw a tree diagram to illustrate the different possibilities c)in hoe many ways will the 2nd , 3rd, and 4th coins all turn heads d)would the same possibilities arise if one coin is tossed 5 times in succession. 2)Det

Discrete structures in mathematics and computer science

Q1) Use the standard logical equivalences to simplify the expression (ㄱp ^ q) v ㄱ(pVq) Q2) consider the following theorem "The square of every odd natural number is again an odd number" What is the hypothesis of the theorem? what is the conclusion? give a direct proof of the theorem. Q3) consider the follo

Mean, Median, Mode and Standard Deviaton

1. Suppose you have administered a test of manual dexterity to two groups of 10 semi-skilled workers. one of these two groups of workers will be employed by you to work in a ware house with many fragile items. the higher the manual dexterity of a worker the less likelyhood that worker wil break significant inventory.Because of a

Discrete Math : Logic (40 MC Problems)

1. Identify the rule of inference used in the following: If it rains today, the flood gates will open. The flood gates did not open today. Therefore, it did not rain. a. modus tollens b. hypothetical syllogism c. modus ponens d. disjunctive syllogism 2. Identify the rule of inference used in the following: If I work all

Discrete Structures Questions

1. By using the Pigeonhole Principle, we can show that if you take six classes in a term and classes do not meet on the weekend, then at least three of the classes must meet on the same day. True False 2. By using the Pigeonhole Principle, it can be shown that if you are paid bi-weekly (every two weeks) duri

RSA encryptions

(See attached file for full problem description) --- Consider the RSA encryption system given by p=43,q=59, and e=13 i) Find d such that ed ≡ 1 (mod (p-1)(q-1)) ii) Decode the message : 1552 2069 1178 1637 1975 Using the convention A = 00, B = 01, ..., Z = 25 ---

Formula - Several Problems

(See attached file for full problem description with proper symbols) --- 2. Let f(x) = x2 +1 and g(x) = {x+1, x> =3; x-1, x<3 so both f and g map R into Find the formula for a. (f+g)(x) b. (f .g)(x) c. (f o g)(x) d. (g o f)(x) 3. Let A = {a,b,c,d} and B = {1,2,3} and let f : A &#61664; B be a function . Let g : Z

Discrete structures located

We worked on the attached problems today in class I am now trying to work through them again for understanding and I am not getting very far. My skills in discrete mathematics are not such that I can work through these on my own effectively. 3. Seven points are located in a plane. List the possible numbers of lines determi

Discrete Structures : Sequences, Subsequences and Remainders

1. How do I prove that in a group of n people there are two people with the same number of acquaintances within the group? 2. Prove that given a sequence of twelve integers, a1, a2, ...,a12, there is a subsequence aj, aj+1, ..., ak where 12 divides &#8721;kn= aa n. 3. A scrape of paper is found in an old desk that read:

Suppose that only 25% of all drivers...

48. Suppose that only 25% of all drivers come to a complete stop at an intersection having flashing red lights in all directions when no other cars are visible... (see attachment for complete question)


On the first day of math class, 20 people are present in the room. To become acquainted with one another, each person shakes hands just once with everyone else. How many handshakes take place?

Multinomial experiment, Expected Frequency

1. True or False. In a multinomial experiment, all outcomes of each trial can have several categories. 2. In finding Expected Frequency you multiply what divided by what?

Discrete Structures : Congruences

Construct addition and multiplication tables for arithmetic modulo 11. For example, 7 + 8 mod 11 is 4 and 7*8 mod 11 is 1 so the entry in row 7, column 8 would be 4 for the addition table and 1 for the multiplication table. Use your tables to solve each of the following congruences: a. 3x+2&#8801;8 (mod11) b. 3x-5&#8801;2 (m


Let G be a complete graph on n vertices. Please calculate how many spanning and induced subgroups G has... (see attachment)