Prove the case involving ∨E of the inductive step of the (strong) soundness theorem for natural deduction in classical propositional logic.

Hint: you need to simultaneously consider 3 different instances of entailment, 1 regular and 2 featuring the transformation of an assumption into a premise.

Show that the set ℝ2R2, equipped with operations

(?1,?1)+˜(?2,?2)=(?1+?2+1,?1+?2−1)(x1,y1)+~(x2,y2)=(x1+x2+1,y1+y2−1)

? ⋅˜ (?,?)=(??+?−1,??−?+1)

(1)defines a vector space over ℝR.

(2)Show that the vector space ?V defined in question 1 is isomorphic to ℝ2R2 equipped with its usual vector space operations. This means you need to define an invertible linear map ?:?→ℝ2T:V→R2.

Classify each function as injective, surjective, bijective, or none of these.

a) f₁ : N --> Q defined by f₁ (n) = n/(n+1)

b) f₂: Z --> Z defined by f₂(n) = n²

c) f₃ : N --> N defined by f₃(n) = n³

d) f₄ : B --> (0, infinity) defined by f₄ (C) = the area of C, where B is the set of all circles in the firs quadrant that are simultaneously tangent to both the positive x and y axes.

e) f₅: A --> (0, infinity) defined by f₅(R) = the area of R, where A is the set of all "not-taller-than-wide" rectangles in the first quadrant with on e angle at the origin (i.e., the one side lying along the positive x-axis and another side lying along the positive y-axis, and the width of any rectangle in A is greater than or equal to its height).

4. (Oblique Trajectory Problem) Let F(x, y) = x 2 + xy + y 2 . Find a formula for G(x, y) such that every curve in the one-parameter family defined by F(x, y) = c intersects every curve in the one-parameter family defined by G(x, y) = c at a sixty degree angle

Problem 10-29 (Algo)

The following table contains the measurements of the key length dimension from a fuel injector. These samples of size five were taken at one-hour intervals. Use three-sigma control limits. Use Exhibit 10.13.

a. Calculate the mean and range for the above samples. (Do not round intermediate calculations. Round your answers to 3 decimal places.)

b. Determine X=X= and R−R− . (Do not round intermediate calculations. Round your answers to 3 decimal places.)

c. Determine the UCL and LCL for a X−X− -chart. (Do not round intermediate calculations. Round your answers to 3 decimal places.)

d. Determine the UCL and LCL for R-chart. (Leave no cells blank - be certain to enter "0" wherever required. Do not round intermediate calculations. Round your answers to 3 decimal places.)

e. What comments can you make about the process?

• Process is in statistical control.

• Process is out of statistical control.

5. (Mixing Problem) A very large tank is initially filled with 100 gallons of water containing 5 pounds of salt. Beginning at time t = 0, a brine solution with a concentration of 1 pound of salt per gallon flows into the top of the tank at 3 gallons per second, the mixture is stirred, and the mixture flows out of the bottom of the tank at 2 gallons per second. (a) Letting w = pounds of salt in the tank at time t seconds, derive a differential equation using the principle dw dt = rate in − rate out. (b) Find a formula for w in terms of t by solving the differential equation in Part (a) after first expressing it in the form P(t, w) dt + Q(t, w) dw =0

The operations manager of a musical instrument distributor feels that demand for a particular type of guitar may be related to the number of YouTube views for a popular music video by the popular rock group Marble Pumpkins during the preceding month. The manager has collected the data shown in the following table: (can be done in excel)

1. Graph the data to see whether a linear equation might describe the relationship between the views on YouTube and guitar sales.
2. Using the equations presented in this chapter, compute the SST, SSE, and SSR. Find the least squares regression line for the data.
3. Using the regression equation, predict guitar sales if there were 40,000 views last month.

3. Suppose that you have 7 cookies to distribute between 13 children.

(a) If Jojo is one of the children, how many ways can you distribute the cookies so that Jojo gets at least two cookies?

(b) If Jojo and Joanne are two of the children, how many ways can you distribute the cookies so that Jojo and Joanne each get at least two cookies?

(c) Answer the above questions for when Jojo, Joanne, and Joey get two cookies each. Also, can we give Jojo, Joanne, Joey, and Josephine two cookies each?

(d) Use what you learned above to determine how many ways at least one of the 13 kids can have at least two cookies.

(e) Use what you learned. above to determine how many ways each of the 13 kids can have at most one cookie.

(f) Answer the question in (e) directly using binomial coefficients. Hint: Think of each kid being a position in a bit string.

(g) What combinatorial identity did you derive in the previous two problems?

The cost in dollars of operating a jet-powered commercial airplane Co is given by the following equation
Co = k*n*v^(3/2)
where
n is the trip length in miles,
v is the velocity in miles per hour, and
k is a constant of proportionality.
It is known that at 590 miles per hour the cost of operation is $300 per mile. The cost of passengers' time in dollars equals $226,000 times the number of hours of travel. The airline company wants to minimize the total cost of a trip which is equal to the cost of operating plus the cost of passengers' time.
At what velocity should the trip be planned to minimize the total cost?
HINT: If you are finding this difficult to solve, arbitrarily choose a number of miles for the trip length, but as you solve it, you should be able to see that the optimal velocity does not depend on the value of n​

2. Consider functions f : {1, 2, 3, 4, 5, 6} → {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

(a) How many of these functions are strictly increasing (i.e. f(1) < f(2) < f(3) < f(4) < f(5) < f(6))? Hint: How many different possibilities are there for the range of f? For each range of f, how many strictly increasing functions are there?

(b) How many of these functions are non-decreasing (i.e. f(1) ≤ f(2) ≤ f(3) ≤ f(4) ≤ f(5) ≤ f(6))? Hint: What are the yards? What are the trees? Or, if you prefer, what are the stars and what are the bars?

Taylor series f(x_i+1)=f(x_i)+f'(x_i)h+(f''(x_i)/2!)h²+(f'''(x_i)/3!)h³+.....

given f''(x_i)=(-f(x_i+3)+4f(x_i+2)-5f(x_i+1​)+2f(x_i))/h²

what is the order of error o(h²)

Using the method of undetermined coefficients determine the exact (only) of a particular solution. Do not evaluate the coefficients.

y''' + 2y'' + y' = 5e^-tsin(t) + 3 + 7te^-t