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# Combinatorics

### Cardinality, Countability and Denumerable Sets

1. Show that if A and B are countable and disjoint, then A U B is countable. 2. Show that any set, A, of cardinality c contains a subset, B, that is denumerable. 3. Show that the irrational numbers have a cardinality c. 4. Show that if A is equivalent to B and C is equivalent to D, then A x C is equivalent to B x D.

### Combinatorial Mathematics : Distinct Elements and Combinations

5. Four numbers are selected from the set: {-5,-4,-3,-2,-1,1,2,3,4} . In how many ways can the selections be made so that the product of the numbers is positive and: a) The numbers are distinct. b) Each number may be selected as many as four times. c) Each number may be selected at most three times.

### Transpositions and Cycles : Let H be a subgroup of Sp

Let H be a subgroup of Sp (the permutation group), where p is prime. Show that if H contains a transposition and a cycle of length p, then H = Sp.

### Functions and Same Cardinality

Find a function that is one-to-one to show the following sets have the same cardinality. Let N=(1,2,3,4,5,6,7,8,9,...) N and A= (2n l n E N)

### Discrete Math - Definitions : Combinatorics, Enumeration

On the following terms could you please give my an English text description - in your own words. Thanks. 1. Combinatorics: 2. Enumeration: 3. Permutation: 4. Relation on A: 5. Rn: 6. Reflexive: 7. Symmetric: 8. Antisymmetric: 9. Transitive:

### Counting Principles and Probability : Combinations, Is the game fair? and Z-score

In a Chinese restaurant, the menu lists 8 items in Column A and 6 items in column B. To order a dinner, the diner is told to select 3 items from column A and 2 from column B. How many dinners are possible? Suppose a family plans 6 children, and the probability that a particular child is a girl is 1/2. Find the probability

### Functions, Enumeration Schemes and Bijection

Let f:Z+ --> Z be the function defined by... f(n) = {n/2 n even { -(n-1)/2 n odd ? n even; The following table indicates the enumeration scheme behind the definiton of the function. n .......... 7 5 312468... f(n)..... ?3 ?2 ?1 0 1 2 3 4... Show that f is a bijection. [You may assume basic facts about the

### Proofs : Denumerable Sets and Cartesian Products

Need help in determining the following proof exercise. (See attached file for full problem description) --- Corollary 14.2.4: If A is a denumerable set then so is A^n for every positive integer n. Proposition 14.2.3: If A and B are denumerable sets then so is their Cartesian product , AX B. ---

### Max and Min Absolute Values

(See attached file for full problem description) --- First: solve this problem. Second: check my answer. Third: if my answer is wrong or incomplete explain why. Find the absolute max and min of on [-8,8] My answer: On the interval [-8,8], f has an absolute max at f(0) = 0 and an absolute min at f(-8) = -14

### Binomial Coefficients and Combinations

Four people select a main dish from a menu of 7 items. How many choices are possible: a) If a record is kept of who selected which choice (as a waiter would) b) If who selected which choice is ignored (as a chef would). Analyze this part by the number of different choices made.

### Finding Binomial Coefficients and Combinations

Committees of 5 individuals are to be formed from 8. a) How many committees are there? b) How many committees of 5 can be formed from 8, given that two particular individuals are to be included on the committee? c) How many committees of 5 can be formed from 8, given that two particular individuals are to be excluded f

### Counting measure

Definition: For any E in X, where X is any set, define M(E) = infinity if E is an infinite set, and let M(E) be then number of points in E if E is finite. M is called the counting measure on X. Let f(x) : R -> [0,infinity) f(j) = { a_j , if j in Z, a if j in RZ} ( Z here is counting numbers, R is set of real numbers)

### Encoding

Encoding problem (See attached file for full problem description)

### Lebesgue Measurable Set

Let A be a set in R^n, we denote by A + x_o a parallel shift of A by x_o to A + x_o, A + x_o = { x : x = y + x_o, y in A}. Now, if A is a lebesgue measurable then show that 1). x_o + A is also lebesgue measurable 2). m(A) = m(x_o + A) Can someone check my answer and tell me if it is correct or not? My work: s

### Lebesgue Measurable Sets, Compact Sets and Open Sets

If A is lebesgue measurable sets in R^n, bounded, then there is a compact set K_epsilon and an open set for every epsilon > 0 V_epsilon such that K_epsilon is subset of A and A is a subset of V_epsilon and for m(A-K) < epsilon m(V-K) < epsilon.

### Arranging Combinations of Sport League Fixtures

In addition to each team playing every other team in the league once at home and once away, the matches should be arranged such that: (1) each team only has one match per week (2) for any team no two matches should be successively at home or away (3) for any team no two successive matches should be against the same opposing

### Count Rectangles : How many rectangles are in.....

Count Rectangles : How many rectangles are in.....

### Power set and bijection

I have two small problems. I need all the work shown and in the second problem please answer in detail and NOT just yes or no. (See attached file for full problem description)

### Three of eight finalists will be selected and awarded \$5,000

Three of eight finalists will be selected and awarded \$5,000.00 scholarships. In how many ways can this selection be made?

### Combinations and Permutations : Ways to paint the 10 identical rooms 5 colors; Find the egf for the number of ways to distribute r distinct objects into n boxes.

Exercise #1 A) How many ways are there to paint the 10 identical rooms in a hotel with five colors if at most three rooms can be painted green, at most three painted blue, at most three red, and no constraint on the other two colors, black and white. B) Show that ...is the generating function for the number of ways a sum o

### Combinations Application Word Problem

Suppose that 30 different computer games and 20 different toys are to be distributed among 3 different bags of Christmas presents. The first bag is to have 20 of the computer games. The second bag is to have 15 toys. The third bag is to have 15 presents, any mixture of games and toys. How many ways are there to distribute these

### Combination Application Word Problems and Summations

Exercise # 1 A) How many nonnegative integer solutions are there to the pair of equations: And B) How many ways are there to distribute 20 toys to m children such that the first two children get the same number of toys if: 1- The toys are identical? 2- The toys are distinct? Exercise #2

### Combinations Application Word Problem

A person has 3 different letters to write, 2 interviews to do, and 2 commercials to review. In making aschedule, (first, second, etc.) how many different combinations are there?

### Golf Scheduling Problem : Combinations and Groups

I have 12 golfers total, and we are playing 6 rounds of golf with 4 golfers in each group. I want to try and have everybody play with each person at least once.

### Example of a Combinations Problem

A disk jockey can play 7 songs during his program. If there are 13 songs to chose from, in how many different orders can the song be arranged?

### How to Find the Possible Number of Codes

How many four letter codes can be formed using A,B,C,D,E and F? No letter can be used more than once.

### Question regarding combinations

If you have six pairs of jeans, three shirts and two pairs of sandals, how many different outfits can you wear?

### Random walk

Hi I would appreciate if you could help me with this question. Is the question below TRUE or FALSE and explain why ? Question: I have two random walks, both starting at 0 and with a reflecting boundary at 0. Each Step, Walk A goes up 1 with probability 1/2 and down 1 with probability 1/2(except at the boundary). Each

### Word problem - A person has 14 close friends.

A person has 14 close friends. (a) Suppose that two of her friends (of the 14 of either gender) do not like each other. If one of the two is invited, the other will not come to the party. How many ways are there to invite 8 people to the party. Explain. (b) Suppose that two of her friends are a couple. She cannot in

### Sequences : Cantor Sets (2 Problems)

1) a. Let with . If is a measurable subset of R, prove that and are measurable. b. let E =[0,1]Q. Prove that E is measurable and (E)=1. c. let P denote the cantor set in [0,1]. Prove that 2) If E R, prove that ther exists a sequence { } of open sets with E for all n such that = . -