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

there are three brands you can choose; H&M, Banana, and Zara. H&M and Banana clothes cost 100 dollars each and Zara clothes cost 200 dollars each.

How many different purchases can you make if you have to spend 1500 dollars and

must buy AT LEAST (NOT EQUAL) as many H&M clothes as Banana clothes?

1) Use dimensional analysis to find a relationship between the force of the wind, F, on a car. You will also need the velocity, v, the surface area of the car, A and the density of air, p.
.
2) Consider the problem of determining the terminal velocity of a raindrop falling from a motionless cloud. Determine a general model using dimensional analysis. Hint: You will need 5 parameters

Let M be the integer corresponding to the first letter of your last name. For example, if your last name begins with "A", M=1 and if your last name begins with "Z", M=26.

Let k=1/M and consider the logistic equation dy/dt = k y (M - y).

Construct a single figure including

1. Title "Logistic Equation Slope Field and Euler's Method solutions by FirstName LastName" with your actual first and last names, along the top
2. Labels for the axes
3. a slope field for -3 ≤ t ≤ 3, -M/2 ≤ y ≤ 3M/2
4. Euler's method solutions for initial conditions   y(0)=5M/4, y(0)=M, y(0)=M/2, y(0)=0 and y(0)=-M/4, in each case for -3 ≤ t ≤ 3, h=0.1
5. For each initial condition, a label "y(0) = (value)" but with the actual value, at an appropriate location
6. For each solution curve, a label "y(3) ≈ (Euler's method value)" but with the actual value, or "y(3) = ?", at an appropriate location

Print on 8.5x11 paper in landscape format.

First letter of my last name is P.

there are three brands you can choose; H&M, Banana, and Zara. H&M and Banana clothes cost 100 dollars each and Zara clothes cost 200 dollars each.

How many different purchases can you make if you have to spend 1500 dollars and

must buy AT LEAST (NOT EQUAL) as many H&M clothes as Banana clothes?

Suppose that A is an n × n matrix satisfying A3 - 3A + 2A-3l = 0.

Show that A is invertible by the definition of invertible. (Hint: Review the definition of invertible, and then describe the inverse in terms of the matrix A - you don’t need to know what A is to answer this question.)

Let P be a finite p-group. Show that Φ(P) is the unique normal subgroup of P minimal such that the corresponding factor group is elementry abelian

How many ways can you distribute 1 piece of chocolate, 2 licorice sticks, and 7 pixie sticks to 5 children so that each child has exactly 2 pieces of candy?

Given a matrix that defines a reflection about a line, find an equation for this line.

Given a matrix that defines an orthogonal projection onto a line, find an equation for this line.

Could yall give me an example of these questions. And solve it for me.

A state legislator wishes to survey residents of her district to see what proportion of the electorate is aware of her position on using state funds to pay for abortions. (Round your answers up to the nearest integer.)

(a) What sample size is necessary if the 95% CI for p is to have a width of at most 0.15 irrespective of p?

(b) If the legislator has strong reason to believe that at least 7/8 of the electorate know of her position, how large a sample size would you recommend to maintain a width of at most 0.15?

Solve the following logic problems. Remember, everyone you meet is either a knight or a knave, knights make true statements, and knaves make false statements. Give your reasoning for each problem. p,q,r and T/F truth table format.

You meet two residents, Alex and Bill. They say the following: Alex: I’m a knight. Bill: Alex is a knight, but I’m a knave. Is Alex a knight or a knave? Is Bill a knight or a knave?

You meet Clara and Davis, who are all like: Clara: One of us is a knight and the other is a knave. Davis: Clara is a knave. Is Clara a knight or a knave? Is Davis a knight or a knave?

You meet Edith and Frank, though only Edith speaks. Edith: Both Frank and I are knaves. Is Edith a knight or a knave? Is Frank a knight or a knave? (Note: Frank’s silence gives no indication of his type, but you can figure out from Edith’s statement.)

You meet Gina, Herbert, and Ichabod. Gina: Ichabod is a knave, if and only if I’m a knight. Herbert: Ichabod is a knight, if and only if I’m a knave. Ichabod: I like pudding. Does Ichabod like pudding?

How to transform x^2+xy+y^2+4x+2y=0 into the standard for of an ellipse and finding the vertices of both major and minor axis. Plot points and graph the ellipse.

For my math class I am required to write an essay on the following topic:

Write an essay about the existence and uniqueness of a solution to an ODE( ordinary differential equation) ?

I do not require that the essay be written but can you possibly give me some hints and extra information that I may include in my essay.

a = [3, -4, 7] b = [-6, 9, 8] c = [4, 0, 8] d =[7, 1, 7] e = [3, -5, 2, 1] f =[5, -7, -3, 6] g = [3, -4, 4, 3]
P = Projection of
ex. C = |g|(gf/gf) C = gf/|f|
ex. P g --> f = Cgf = C(gf/f) (1/|f|) (f) =( gf/ff)(f)
Find
a. Pg --> f
b. Pa --> 3b + e

Find (cross multiply)
a. ||a X b||
b. ||g X e||
c. ||d X b||
d. Calculate d X b
e. Calculate c X d

4. The product y = Ax of an m n matrix A times a vector x = (x1; x2; : : : ; xn)T can be computed row-wise as y = [A(1,:)*x; A(2,:)*x; ... ;A(m,:)*x]; that is y(1) = A(1,:)*x y(2) = A(2,:)*x ... y(m) = A(m,:)*x Write a function M-file that takes as input a matrix A and a vector x, and as output gives the product y = Ax by row, as denoted above (Hint: use a for loop to denote each entry of the vector y.) Your M-file should perform a check on the dimensions of the input variables A and x and return a message if the dimensions do not match. Call the file myrowproduct.m. Note that this le will NOT be the same as the myproduct.m example. Test your function on a random 5x2 matrix A and a random 2x1 vector x. Compare the output with A*x. Repeat with a 3x5 matrix and a 5x1 vector and with a 3x5 matrix and a 1x5 vector. Use the command rand to generate the random matrices for testing. Include in your lab report the function M-file and the output obtained by running it.

5. Recall that if A is an m n matrix and B is a p q matrix, then the product C = AB is denoted if and only if n = p, in which case C is an m q matrix.

(a) Write a function M-file that takes as input two matrices A and B, and as output produces the product by columns of the two matrix. For instance, if A is 3x4 and B is 4x5, the product is given by the matrix C = [A*B(:,1), A*B(:,2), A*B(:,3), A*B(:,4), A*B(:,5)]The function file should work for any dimension of A and B and it should perform a
check to see if the dimensions match (Hint: use a for loop to denote each column of C). Call the file columnproduct.m.Test your function on a random 5x3 matrix A and a random 3x5 matrix B . Compare the output with A*B. Repeat with 3 x 6 and 6 x 4 matrices and with 3 x 6 and 4 x 6 matrices.Use the command rand to generate the random matrices for testing. Include in your lab report the function M-file and the output obtained by running it.

(b) Write a function M- file that takes as input two matrices A and B, and as output produces the product by rows of the two matrices. For instance, if A is 3 x 4 and B is 4x5, the product AB is given by the matrix C = [A(1,:)*B; A(2,:)*B; A(3,:)*B] The function file should work for any dimension of A and B and it should perform a check to see if the dimensions match (Hint: use a for loop to denote each row of C). Call the file rowproduct.m. Test your function on a random 5x3 matrix A and a random 3x5 matrix B . Compare the output with A*B. Repeat with 3 x 6 and 6 x 4 matrices and with 3 x 6 and 4 x 6 matrices. Use the command rand to generate the random matrices for testing.
Include in your lab report the function M-file and the output obtained by running it.