I am having the most trouble with 1d:

1. a. Prove that if f : A → B, g : B → C, and g ◦f : A → C is a 1-to-1 surjection, then f is 1-to-1 and g is a surjection.
Proof.
b. Prove that if f : A → B, g : B → C, g ◦ f : A → C is a 1-to-1 surjection, and g is 1-to-1, then f is a surjection.
Proof.
c. Prove that if f : A → B, g : B → C, g ◦ f : A → C is a 1-to-1 surjection, and f is a surjection, then g is 1-to-1.
Proof.
d. Find functions f : R → R and g : R → R such that g◦f is a 1-to-1 surjection , f is not a surjection, and g is not 1-to-1.

Let R and S be rings. Denote the operations in R as +_R and ·_R and the operations in S as +S and ·S

(i) Prove that the cartesian product R × S is a ring, under componentwise addition and multiplication.

(ii) Prove that R × S is a ring with identity if and only if R and S are both rings with identity.

(iii) Prove that R × S is a commutative ring if and only if R and S are both commutative rings.

Suppose that a process at Manuel Automation with a normally distributed output has a mean of 50.0 cm and a variance of 3.61 cm. a. If the specifications are 51.0 ± 3.75 cm, compute Cp, Cpu, Cpl, and C pk. b. Suppose the mean shifts to 52.0, but the variance remains unchanged. Recompute and interpret these process capability indexes. c. If the variance can be reduced to 40 percent of its original value, how do the process capability indices change (using the original mean of 50.0)?

Use Simpson's Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by your calculator. (Round your answer to six decimal places.)

y = x + sqrt x, 2 ≤ x ≤ 5

Solve y'' + 16y = 7cos(4t) using variation of parameters. Then solve using Laplace transformations given y(0) = 1 and y'(0) = 2

can anyone tell me the reason why if the number of basis is equal to the dimension of a vector space V then it is the basis of the vector space V. and also what the theorem is? since, I think if V=span{(1,0,0), (0,1,0)} and dim(V)=2 and basis={(1,0,0), (0,0,1)} which the number is also 2. but it is not the basis of V. So, can you tell where is the mistake. THANK YOU!

y'' + y=tanx+3x-1 solve the ode and find the general solution?

Your firm produces two products, Thyristors (T) and Lozenges (L), that compete for the scarce resources of your distribution system. For the next planning period, your distribution system has available 6,000 person-hours. Proper distribution of each T requires 3 hours and each L requires 2 hours. The profit contributions per unit are 40 and 30 for T and L, respectively. Product line considerations dictate that at least 1 T must be sold for each 2 L’s.

(a) Draw the feasible region and draw the profit line that passes through the optimum point.

(b) What are the constraints for this problem?

(c) By simple common sense arguments, what is the optimal solution?

Use the Laplace transform to solve the problem with initial values

y''-2y'+2y=cost
y(0)=1
y'(0)=0

2. Show that Σ ⊆ Prop(A) is consistent and complete iff there is exactly one truth assignment that satisfifies Σ.

Floyd’s Bumpers has distribution centers in Lafayette, Indiana; Charlotte, North Carolina; Los Angeles, California; Dallas, Texas; and Pittsburgh, Pennsylvania. Each distribution center carries all products sold. Floyd’s customers are auto repair shops and larger auto parts retail stores. You are asked to perform an analysis of the customer assignments to determine which of Floyd’s customers should be assigned to each distribution center. The rule for assigning customers to distribution centers is simple: A customer should be assigned to the closest center. The worksheet Floyds in the provided datafile contains the distance from each of Floyd’s 1,029 customers to each of the five distribution centers. Your task is to build a list that tells which distribution center should serve each customer. The following functions will be helpful:

=MIN(array).

The MIN function returns the smallest value in a set of numbers. For example, if the range A1:A3 contains the values 6, 25, and 38, then the formula =MIN(A1:A3) returns the number 6, because it is the smallest of the three numbers:

=MATCH(lookup_value, lookup_array, match type).

The MATCH function searches for a specified item in a range of cells and returns the relative position of that item in the range. The lookup_value is the value to match, the lookup_array is the range of search, and match type indicates the type of match (use 0 for an exact match).

For example, if the range A1:A3 contains the values 6, 25, and 38, then the formula =MATCH(25,A1:A3,0) returns the number 2, because 25 is the second item in the range.

=INDEX(array, column_num).

The INDEX function returns the value of an element in a position of an array. For example, if the range A1:A3 contains the values 6, 25, and 38, then the formula =INDEX(A1:A3, 2) 5 25, because 25 is the value in the second position of the array A1:A3. (Hint: Create three new columns. In the first column, use the MIN function to calculate the minimum distance for the customer in that row. In the second column use the MATCH function to find the position of the minimum distance. In the third column, use the position in the previous column with the INDEX function referencing the row of distribution center names to find the name of the distribution center that should service that customer.)

Click on the datafile logo to reference the data.

(Hint: The INDEX function may be used with a two-dimensional array: =INDEX(array, row_num, column_num), where array is a matrix, row_num is the row numbers and column_num is the column position of the desired element of the matrix.)

Floyd's Bumpers pays a transportation company to ship its product to its customers. Floyd's Bumpers ships full truckloads to its customers. Therefore, the cost for shipping is a function of the distance traveled and a fuel surcharge (also on a per mile basis). The cost per mile is $2.54 and the fuel surcharge is $.56 per mile. The worksheet May in the provided datafile contains data for shipments for the month of May (each record is simply the customer zip code for a given truckload shipment), as well as the distance table from the distribution centers to each customer. Use the VLOOKUP function to retrieve the distance traveled for each shipment from the exercise completed above, and calculate the charge for each shipment. What is the total amount that Floyd's Bumpers spends on these May shipments?

If required, round your answers to two decimal places.

Given the following subsets of R:

A = R\Q = {x ∈ R|x not in Q}

B = {1, 2, 3, 4}

C = (0, 1]

D = (0, 1] ∪ [2, 3) ∪ (4, 5] ∪ [6, 7] ∪ {8}

(a) Find the set of limit points for each subset when considered as subsets of RU (usual topology on R).

(b) Find the set of limit points for each subset when considered as subsets of RRR (right-ray topology on R).

You are a famous archaeologist/treasure hunter ́a la Indiana Jones. After following a treasure map you find yourself deep inside a Babylonian temple. As you reach the end of a long corridor you find it splits into two paths. Above the first path you make out some text carved into the rock. Shining your torch you manage to make out the following two inscriptions on the wall:

1. L1 ∧ T2
2. (L1 ∧T2)∨(L2 ∧T1)

Knowing that the Babylonians were great mathematicians, you’re not surprised to see that they had developed such a refined system of propositional logic centuries before it should have been. The historians of mathematics will surely want to hear of this discovery when you’re done!

Having no idea what these variables could mean however, you look down at your map to see if there are any hints. You notice scrawled in the margins of the map “L1: First Path Leads to Being Lost Forever”, “L2: Second Path Leads to Being Lost Forever”, “T1: First Path Leads To Treasure” and “T2: Second Path Leads To Treasure”.

Reading the first inscription you quickly translate the treasure is down the second path. However, as you’re about to step into the tunnel you remember something the map seller said as you were leaving his shop: “One tells the truth and the other is a lie!” You had thought that was cryptic nonsense at the time but thank goodness you remembered! He seemed like a trustworthy guy so you’ll assume that his statement was true and that one inscription is lying and the other is telling the truth.

Q: (20 points) Assuming the lying inscription is true and the truthful inscription is false leads to a contradiction. Prove this using the laws of propositional logic. First, combine the two statements into a single boolean expression (adding a ¬ to the expression that you’re assuming is false). Then proceed using the laws of propositional logic to arrive at “False”. You must show each step and identify which law you are applying. You must use the distributive law at least once; we are looking for you to demonstrate mastery over several laws rather than a quick solution.

A = [4, 5, 9]
B = [-4, 5, -7]
C = [2, -7, -8, 5]
D = [1, -9, 5, -3]
E = [3, 3, -1]

Uz = 1/|z| ^z
d(X,Y) = (Rθ) d = diameter R = Radius θ = Theta

Find

a. Uc
b. d (D, C)
c. Let P = B + 3E, UP =
d. A x B
e. 3B x E
f. C x D