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Discrete Math

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 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∩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

Algorithm

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.

Proof that a sequence is monotone increasing

(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 . ---

Correspondence of Borel sets

If f is one-to-one, f, f^-1 are continuous, then f is called a homeomorphism. Now I want you to prove the following: Let f : X -> Y, ( X and Y are topological spaces)be homeomorphism, prove that it establishes one-to-one correspondence between Borel sets in X and Y.

Proof Limit Solved

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))+

Solving a recurrence

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!

Irreducible representation proofs

1. Assume that the field is algebraically closed and has zero characteristic, G is finite and representations are finite-dimensional. Show that this statement is true under the above assumptions: "Let p be an irreducible representation of G, and q be an irreducible representation of H. Is it always true that the exterior t

Calculating Time from the Number of Loops

How much time does the following algorithm require as a function of n? Express your answer in theta notation in the simplest form. Consider each individual instruction (including loop control) as elementary. l = 0 for i = 1 to n for j = 1 to n^2 for k = 1 to n^3 l = l + 1

Proof of Differentiability

Let f(x) = { x^2 if x is rational { 0 if x is irrational Show that f is differentiable at x=0 but not at any other point. --- Please see the attached file for the fully formatted problem.

Sets of Six, Seven and Eight-Letter Words

(a) Give an example of a set S such that the language S* has more six-letter words than seven-letter words. (b) Give an example of a set S such that the language S* has more six-letter words than eight-letter words. (c) Does there exist a set S such that S* contains more six-letter words than twelve-letter words. Give reas

5 Finite Math Problems

1) A man walking in the woods encounters a stream. Because he is unsure of the stream depth, he measures how deep the water is in many random spots along the entire width of the stream. After 1000 measurements each with a depth of 6 inches, he concludes the probability of the water being 6 inches deep the entire way across to

Finite Math

1.) Dan borrows $500 at 9% per annum simple interest for 1 year and 2 months. What is the interest charged, and what is the amount due? ________________________________________ 2.) A mutual fund pays 9% per annum compounded monthly. How much should I invest now so that 2 years from now I will have $100 in the account?

Transitive Closures

I've attached the problem I'm having trouble with. Please provide assistance. (See attached file for full problem description).

Removable singularity

(See attached file for full problem description) --- Let have an isolated singularity at and suppose that is bounded in some punctured neighborhood of . Prove directly from the integral formula for the Laurent coefficients that for all j = 1,2,3,..., i.e. must have a removable singularity at . The integ

Finite Math

(See attached file for full problem description) --- 1. What is marginal cost? Fixed cost? Find the slope for each line that has a slope. 3. Through (-2, 5) and (4, 7) 5. Through the origin (11, -2) 7. 2x + 3y = 15 9. y + 4 = 9 11. y = -3x Find an equation in the form y = mx + b (where possible) for each line.