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Permutations as Product of Disjoint Cycles

Modern Algebra Group Theory (LXXIX) Permutation Groups The Orbits and Cycles of Permutations Write the given permutation as the product of disjoint cycles 1 2 3 4 5 6 6 5 4 3 1 2 The fully formatted problem is in the attached file.

Functions and countable sets

(See attached file for full problem description with all symbols) --- 2.14 (I) Prove that an infinite set X is countable if and only if there is a sequence of all the elements of X which has no repetitions. (II) Prove that every subset S of a countable set X is itself countable. (III) Prove that if

Solutions Modulo congruences

In order to solve the congruence 2x + 6 ≡ 4 (mod 8), your friend Phil Lovett wrote down the following steps: 2x+6 ≡ 4 (mod 8) x+3 ≡ 2 (mod8) x ≡ −1 (mod 8) From here, Phil concludes that the solution set to 2x + 6 ≡ 4 (mod 8) is {x; x ≡ −1 (mod 8)}. (a) Is Phil's

Combinations of Integers

Show that if 7 integers are selected from the first 10 positive integers there must be at least 2 pairs of these integers with the sum 11 Is the conclusion true if 6 integers are selected instead of 7 How many numbers must be selected from the set {1,3,5,7,9,11,13,15} to guarantee that at least one pair of these numbers a


1. how many license plates can be made using three letters followed by the three digits or four letters followed by two digits 2. how many bit strings of length 10 contain either 5 consecutive 0s or five consecutive 1s.

Sets and Set Operations : Union and Intersection

Create two sets. Set A will be the list of the five items you personally need to buy the most (essential items). Set B will be the list of the five items that you want to buy the most (fun stuff). List the items in Set A and Set B, and also list the items in the unions and intersections of Set A and Set B. Now assume that the p

Proofs with Sets

(See attached file for full problem description). a) Determine the irreducibility of x20-11 over Q(set of rationals), and use it to prove or disprove that the ideal <x20-11> is a maximal ideal of Q[x]. b) Construct an integral domain R and an element a in R such that a is irreducible but not prime in R. c) Suppose that R is

Combinations for Test Answers

A teacher makes 2 tests that each have: -5 multiple choice questions with 4 possible answers -10 true/false questions (2 possible answers) If a students takes both tests. How many possible combinations of answers are there? Would like to know how to figure this out. Is there a forumula? Does it involve factorials?

Combinations and Permutations

11. A computer lab contains the following computers - a Hewlett Packard, a Compaq, a Sony, a Dell and 3 different models of Macs. How many different ways can the 7 computers be arranged so that the Macs are all together? (You may assume the computers are all in one line.) 12. A public pool employs 17 lifeguards of whic

Counting Problems and Trees

In one residence, cell phones, lap top computers and digital tvs are very popular among students. In fact, all of the students own atleast one of these items, although onlu 15 own all 3. Cell phones are the most popular with twice as many students owning cell phones as own lap tops. and digital tvs are still rare, since only hal

Trees - Counting Problems and Combinations

A Pasta bar has a build your own pasta option. On this option, the menu lists 4 types of pasta noodles, 6 different vegetable choices, 4 meat choices, 3 cheeses and 3 sauces. Each customer who orders this option must choose 1 type of noodle and 1 type of sauce. The customer can choose as many of the vegetables as desired and up

Permutations for Different Cycles

Some boys come into a room and sit in a circle. Four girls come in and arrange themselves between 2 of the boys, that is , the 4 girls join the circle, but all sit together. If there are 2880 different circles that could be formed in this way, how many boys are there?

Function Example Problem

F(x) = (mx +7)^1/2 where x >= -7/m and m is a positive constant. It is given that y = f(x) and y = f^-1 (x) do not meet. Explain how it can be deduced that neither curve meets the line y = x, and hence determine the set of possible values of m.

Satisfy Specific Properties - Bounded Above or Below and Integer

In each part of this problem, display an example of a set M with the specific property or properties. a) M does not equal R (R is the set of all real numbers), but M is bounded neither above nor below. b) M is bounded above but fails to contain its least upper bound. c) M is the set of integers that contains neither a s

Negative Integers Proof : Denote by -P the set of all negative

2. Denote by -P the set of all negative integers, i.e., the set to which the number m belongs only in case there is a member n of P such that m = -n. If the number m is in -P and Z is a number such that m<Z, then Z is not in P. If the conjecture is true, prove it. If it's false, prove that its false by counterexample or a proo

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.

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)