### Discrete mathematics

Find a transitive closure of the relation R on {a,b,c,d,e} given by R= {(a,b), (a,c), (a,e),(b,a), (b,c),(c,a), (c,b),(d,a,),(e,d)}

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Find a transitive closure of the relation R on {a,b,c,d,e} given by R= {(a,b), (a,c), (a,e),(b,a), (b,c),(c,a), (c,b),(d,a,),(e,d)}

Show that the symmetric closure of the union of 2 relations is the union of their symmetric closures.

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.

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

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

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

Modern Algebra Set Theory (III) Equivalence Classes of an Equivalence Relation Let ~ be an equi

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

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

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

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

(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

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.

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 ≥ 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

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

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.

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

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

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 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

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∩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

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.

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

(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 pro

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

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

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.

(See attached file for full problem description and equations) --- Prove that the sequence is monotone increasing. Use the following hints: 1) If ln f(x) is increasing, then so is f(x). 2) If , then f is increasing. 3) ln x is defined to be . ---

Criticize the following proof...(see attachment)

I received the following proof, can someone show all steps of how the solution was formed? Proof: Let n=2^2^k, then we have T(n)=T(2^2^k)=2T(n^(1/2))+log n ***How do you get n^(1/2) equals 2^2^(k-1) =2T(2^2^(k-1))+2^k ***How do you get 2T(2^2^(k-1)) equals 2(2T(2^2^(k-2))+2^(k-1)) =2(2T(2^2^(k-2))+2^(k-1))+

Solve the following recurrence exactly for n of the form 2^2^k. T(2) = 1 T(n) = 2T(n^(1/2)) + log n Express your answer as simply as possible using theta notation. note added ** theta notation is based on big O notation Show all work!