##### Let (G,+) be an abelian group and U a subgroup of G. Prove that G is...

Let (G,+) be an abelian group and U a subgroup of G. Prove that G is the direct product of U and V (where V a subgroup of G) if only if there is a homomorphism f : G → U with    f|U = IdU

##### Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer...

Boise Lumber has decided to enter the lucrative prefabricated housing business. Initially, it plans to offer three models: standard, deluxe, and luxury. Each house is prefabricated and partially assembled in the factory, and the final assembly is completed on site. The dollar amount of building material required, the amount of labor required in the factory for prefabrication and partial assembly, the amount of on-site labor required, and the profit per unit are as follows.

Standard Model Deluxe Model Luxury Model
Material $6,000$8,000 $10,000 Factory Labor (hr) 240 220 200 On-Site Labor (hr) 180 210 300 Profit$3,400 $4,000$5,000

For the first year's production, a sum of \$8,200,000 is budgeted for the building material; the number of labor-hours available for work in the factory (for prefabrication and partial assembly) is not to exceed 215,000 hr; and the amount of labor for on-site work is to be less than or equal to 234,000 labor-hours. Determine how many houses of each type Boise should produce to maximize its profit from this new venture. (Market research has confirmed that there should be no problems with sales.)

 standard model houses deluxe model houses luxury model houses

##### Let R be a UFD and let F be a field of fractions for R. If...

Let R be a UFD and let F be a field of fractions for R. If f(α) = 0, where fR [x] is monic and αF, show that αR

NOTE: A corollary is the fact that mZ and m is not an nth power in Z, then n√m is irrational.

##### The​ input-output matrix for a simplified economy with just three sectors​ (agriculture, manufacturing, and​ households) is...

The​ input-output matrix for a simplified economy with just three sectors​ (agriculture, manufacturing, and​ households) is given below.

Agriculture Manufacturing Households

 Agriculture Manufacturing Households
 0.29 0.44 0.105 0.12 0.11 0.115 0.76 3.53 0.123

A.  How many units from each sector does the agriculture sector require to produce 1​ unit?

The agriculture sector requires _____units from​ agriculture, ____units from​ manufacturing, and ___units from households.

B.  What production levels are needed to meet a demand of 32 units of​ agriculture, 34 units of​ manufacturing, and 34 units of​ households?

Production levels of ____units of​ agriculture,___units of​ manufacturing, and _____units of households are needed.  ​(Round to the nearest whole number as​ needed.)

C.  How many units of manufacturing are used up in the​ economy's production​ process?

The​ economy's production process uses up_____ units of manufacturing.

​(Round to the nearest whole number as​ needed.)

##### Previously, we listed all 29 topologies on the set X={a,b,c}. However, some of the resulting topological...

Previously, we listed all 29 topologies on the set X={a,b,c}. However, some of the resulting topological spaces are homeomorphic. Which are homeomorphic? Divide the set of 29 topological spaces into homeomorphism classes, and be sure to justify your choices. There are 9 homeomorphism classes in total. (To justify your choices, explain why the spaces within each class are homeomorphic to each other. Your explanations can be somewhat loose).

##### Suppose that a decision-maker’s preferences over the set A={a, b, c} are represented by the payoff...

Suppose that a decision-maker’s preferences over the set A={a, b, c} are represented by the payoff function u for which u(a) = 0, u(b) = 1, and u(c) = 4.

(a) Are they also represented by the function v for which v(a) =−1, v(b) = 0, and v(c) = 2?

(b) How about the function w for which w(a) =w(b) = 0 and w(c) = 8?

(c) Give another example of a function f:A→R that represents the decision-maker’s preferences.

(d) Is there a function that represents the decision-maker’s preferences and assigns negative numbers to all elements of A?

##### Let G = Z4 × Z4, H = ⟨([2]4, [3]4)⟩. (a) Find a,b,c,d∈G so that G...

Let G = Z4 × Z4, H = ⟨([2]4, [3]4)⟩.
(a) Find a,b,c,d∈G so that G is the disjoint union of the 4 cosets a+H,b+
H, c + H, d + H. List the elements of each coset.
(b) Is G/H cyclic?

##### please solve it step by step 12. Find the angle they form: a) The correct ?1:...

please solve it step by step
12. Find the angle they form:
a) The correct ?1: ? = −1 + 2?, ? = −?, ? = 2 + 4? and
?2: ? = ?, ? = −1 − ?, ? = 4 + 2?
b) Losplanos2? + 3? − ? = 0; ? − ? − ? = 2
c) Larecta?1: ? = 1 + 2?, ? = 2 − ?, ? = −3−3? yelplano ?: ? + ? + ? − 10 = 0

##### Let u and v be orthogonal vectors in R3 and let w = 3u + 6v....

Let u and v be orthogonal vectors in R3 and let w = 3u + 6v. Suppose that ||u|| = 5 and ||v|| = 4. Find the cosine of the angle between w and v.

##### Use Theon's double sequence (Xn and Yn) method to find the first 10 rational approximations to...

Use Theon's double sequence (Xn and Yn) method to find the first 10 rational approximations to square root of 11. You may use a computer program

##### A topological space X is zero − dimensional if it has a basis B consisting of...

A topological space X is zero − dimensional if it has a basis B consisting of open sets which are simultaneously closed. (a) Prove that the set C = {0, 1}N with the product topology is zero-dimensional. (b) Prove that if (X, d) is a metric space for which |X| < |R|, that is the cardinality of X is less than that of R, then X is zero-dimensional.

##### Suppose when searching for a large prime, our first step is to sieve by eliminating the...

Suppose when searching for a large prime, our first step is to sieve by eliminating the first n primes. How large should n be to speed up our search by a factor of 10? This requires careful thought and inventiveness before doing a computation.

How to approach this: Eliminating all odd numbers speeds up our program by a factor of 2, since we've eliminated half of the choices that are composite. If we also immediately disqualify multiples of 3, then we speed up our program by a factor of 3, since we have eliminated 2/3 of all composites. Check that that argument is correct. Now see what happens when you eliminate all of 2, 3 and 5. Again for 2,3,5,7. Now try to figure out the mathematical result and use it to solve for n.

##### Are the following statements true or false? 1. Let P(n) be the statement "If any string...

Are the following statements true or false?

1. Let P(n) be the statement "If any string of length n over {a, b} has more a's than b's, then it has two a's in a row". We can prove this statement is true for all n with n ≥ 2 by proving P(2), P(3), and "for all n: P(n) → P(n+2)".

2. Let P(x) be a predicate with one free variable x of type natural. If I prove P(0), "for all x: P(x) → P(x+2)", and "for all x: P(x) → P(x+3)", I may conclude "for all x: P(x)".

##### 2. An industrial sewing machine uses ball bearings that are targeted to have a diameter of...

2. An industrial sewing machine uses ball bearings that are targeted to have a diameter of 0.75 inches. The lower and upper specification limits under which the ball bearings can operate properly are 0.74 inches and 0.76 inches, respectively.

Past experience has indicated that the actual diameter of the ball bearings is approximately normally distributed with a mean of 0.753 inches and a standard deviation of 0.004 inches.

What is the probability that a ball bearing is:

a. between the target and the actual mean?

b. between the lower specification limit and the target?

c. above the upper specification limit?

d. below the lower specification limit?

e. Of all the ball bearings, 93% of the diameters are greater than what value?

• Use Excel to calculate Z-scores
• Use Appendix Table B.3 Areas Under the Normal Curve from your text to find table probabilities.
• Write complete sentences to answer each question. “The probability of …. is …..”