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# Discrete Math

### Set Theory - : Commutative Laws : Prove that AUB = BUA

Modern Algebra Set Theory (VIII) Laws of Algebra of Sets

### 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

### If 'B is subset of A' prove that 'A union B = A' and convers

Modern Algebra Set Theory (V) Laws of Algebra of Sets

### Equivalence Classes of an Equivalence Relation

Modern Algebra Set Theory (II) Equivalence Classes of an Equivalence Relation The distinct e

### Set Theory - Equivalence Relation

Modern Algebra Set Theory (I) Equivalence Relation Let S be the set of all i

### Discrete Math : Numbers of Solutions to Equations

A) Let n and r be positive integers. Explain why the number of solutions of the equation x1 + x2 + ... + xn = r, where xi is a nonnegative integer for i = 1, 2, 3, ..., n, equals the number of r-combinations of a set with n elements. b) How many solutions in nonnegative integers are there to the equation x1 + x2 + x3 + x4 = 17

### 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 Mathematics and its Applications

22. Prove or disprove then n^2 - 1 is composite whenever n is a positive integer greater than 1.

### 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

### Discrete Math : History, Appearance and Application of Chinese Remainder Theorem in Chinese and Hindu Writings

1. Describe the history of the Chinese Remainder Theorem. Describe some of the relevant problems posed in Chinese and Hindu writings and how the Chinese Remainder Theorem applies to them. Please show references.

### 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

Discuss how an axiomatic set theory can be developed to avoid Russell's paradox. Use the internet as a source for writing a one or two page essay on the topic. Please site your sources. Thanks.

### Discrete Mathematics and its Applications

For each of these sentences, determine whether an inclusive or an exclusive or is intended. Explain your answer. a) Experience with C++ or Java is required. b) Lunch includes soup or salad. c) To enter the country you need a passport or a voter registration card. d) Publish or perish

### 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

### Discussing the Proof

Need help in determining the following proof. (See attached file for full problem description) --- Thm 11.1.2 (the pigeonhole principle): Suppose that f:X&#61664; Y is a function between non-empty finite sets such that |X| > |Y|. Then f is not an injection, i.e. there exist distinct elements x1 and x2 E (epsilon) X

### 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

### Set Theory Proofs: Addition Principle

Let X and Y be finite sets. a) Suppose the X C Y and |X| = |Y|. Use 10.2.1 to prove X =Y. b)... Theorem 10.2.1 (The adddition principle): Suppose that X and Y are disjoint finite sets. Then X U Y is finite and | X U Y| = |X| + |Y|. Please see the attached file for the fully formatted problems.

### Induction & Set Theory: Union & Pairwise Disjoint Finite Sets

Proposition 10.2.1: (the addition principle) Suppose that X and Y are disjoint finite sets. Then X U Y is finite and |X U Y| = |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 X

### Orthonormal Set

(See attached file for full problem description with proper symbols) Assume that is a linearly independent set in a Hilbert space Suppose that is an orthonormal set in satisfying the following property: for each (a) Show that for each (b) Let be the orthonormal set gotten from the Gram-Schmidt procedur

### 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

### Proof: Tree Contains a Cycle

Prove that a graph with n nodes and more than n-1 edges must contain at least one cycle.

### Logic problem and divisors

Suppose you have 2 pitchers... --- (See attached file for full problem description)