Let H and K be subgroups of a group G so that for all h in H and k in K there is a k' in K with hk = k'h. Proposition 2.3.2 shows that HK is a group. Show that K is a normal subgroup of HK.

Let a_n denote the number of different ways to color the walls of a five-sided room with n colors if you insist that two walls that meet at a corner must be assigned different colors.

(i) compute a1, a2 and a3 directly

(ii) Find the formula for a_n

Let A = {1,2,3}. Determine all the equivalence relations R on A. For each of these, list all ordered pairs in the relation.

Using induction: Show that player 1 can always win a Nim game in which the number of heaps with an odd number of coins is odd.

Use the branch and bound method to find the optimal solution to the following integer programming problem: maximize 7x1 + 3x2 subject to: 2 x1 + x2 < 9 3 x1 + 2x2 <13 x1, x2 > 0; x1, x2 integer
Instead of using EXCEL Solver to solve this problem directly as an integer programming problem, use EXCEL Solver to solve the LP problems at each branch, with the appropriate constraints added, according to the branch and bound algorithm. Be sure to draw a node and branch diagram to illustrate the procedure, showing the branches that are fathomed, noninteger variables selected, and the optimal solution

Maximize p = x + 8y subject to

P=? (X,Y)= ? (NOT BY GRAPHING)

WEB ANALYTICS

Write a review applying some of the concepts to the fashion industry. Offer an introduction and a summary, Conduct additional research and offer at least four additional references and citations. Offer references and at least one citation for every reference.

For any Gaussian Integer z ∈ ℤ[i] with z = a+bi , define N(z) =a² + b². Using the division algorithm for the Gaussian Integers, we have show that there is at least one pair of Gaussian integers q and r such that w = qz + r with N(r) < N(z).

(a) Assuming z does not divide w, show that there are always two such pairs.

(b) Fine Gaussian integers z and w such that there are four pairs of q and r that satisfy the division algorithm with N(r) < N(z).

Definition 1 (Topological space). Let X be a set. A collection O of subsets of X is called a topology on the set X if the following properties are satisfied:

(1) emptyset ∈ O and X ∈ O.

(2) For all A,B ∈ O, we have A∩B ∈ O (stability under intersection).

(3) For all index sets I, and for all collections {Ui}i∈I of elements of O (i.e., Ui ∈ O for all i ∈ I), we have U i∈I Ui ∈ O (stability under arbitrary unions). A set X equipped with a topology O is called a topological space and the sets in O are called open sets.

Exercise 1. Let X be a set. (1) Consider O_trivial = {emptyset,X}. Prove that O_trivial is a topology on X. (2) Consider O_discrete = P(X). Is O_discrete is a topology on X? Justify briefly your answer. Hint. You have to verify whether the collections O_trivial and O_discrete satisfy the three properties in Definition 1.

Problem 4 (Sets defined inductively) [30 marks] Consider the set S of strings over the alphabet {a, b} defined inductively as follows: • Base case: the empty word λ and the word a belong to S • Inductive rule: if ω is a string of S then both ω b and ω b a belong to S as well. 1. Prove that if a string ω belongs to S, then ω does not have two or more consecutive a’s. 2. Prove that for any n ≥ 0, if ω is a string of length n over the alphabet {a, b} that does not have two or more consecutive a’s, then ω is a string of S.

Find the break-even point for the revenue and cost. Draw your own graph on a piece of paper.

Fixed cost = $22,000
Variables cost = $15 per item
Price = $23

The break-even point is at ________ items.

Use the data in the following table to create a fraction nonconforming (p) chart. The column of np represents the number of non-conforming units. Is the process in control? (5 points)

Question 2

A bank's manager has videotaped 20 different teller transactions to observe the number of mistakes being made. Ten transactions had no mistakes, five had one mistake and five had two mistakes. Compute proper control limits at the 90% confidence level. Is the process in control? Show your work.

Find the series solution to the following differential equation:

y" + xy' - y = x^2 - 2x , about x = 1

Let x, y ∈ R. Prove the following:

(a) 0 < 1

(b) For all n ∈ N, if 0 < x < y, then x^n < y^n.

(c) |x · y| = |x| · |y|