The goal is to show that a nonempty subset C⊆R is closed iff there is a continuous function g:R→R such that C=g−1(0).
1) Show the IF part. (Hint: explain why the inverse image of a closed set is closed.)
2) Show the ONLY IF part. (Hint: you may cite parts of Exercise 4.3.12 if needed.)

Let the surface (S) be the part of the elliptic paraboloid z = x2 + 4y2lying below the plane z = 1. We define the orientation of (S) by taking the unit normal vector ⃗n pointing in the positive direction of z− axis (the inner normal vector to the surface). Further, let C denotes the curve of the intersection of the paraboloid z = x2 + 4y2 and the plane z = 1 oriented counterclockwise when viewed from positive z− axis above the plane and let S1 denotes the part of the plane z = 1 inside the paraboloid z = x2 +4y2 oriented upward.

a) Parametrize the curve C and use the parametrization to evaluate the line integral

?

F· d⃗r,C

where F(x, y, z) = 〈y, −xz, xz2〉.

b) Find G = ∇ × F, where F(x,y,z) is the vector field from Part a), parameterize the surface S1 and use the parametrization to evaluate the flux of the vector field G.
HINT: The area enclosed by an ellipse x2 + y2 = 1 is abπ.

c) What is the flux of the vector field G = ∇ × F, from Part b), across the surface (S)? Explain why the answers in a), b), and c) must be the same.

Show if A, B, C are connected subsets of X and A∩B not equal ∅ and B ∩C not equal ∅, then A∪B ∪C is connected.

The crossing number of a simple graph is the minimum number of crossings that can occur when this graph is drawn in the plane, where no three curves representing edges are permitted to cross at the same point. Find the crossing numbers of (a) K3,3 (b) K5.

The Ostrowski method for finding a single root of ?(?)=0 is given by
Initial guess ?0 ??=??−?(??)?′(??), ??+1=??−?(??)?′(??) ?(??)?(??)−2?(??).
a) Write MATLAB or OCTAVE coding to implement the Ostrowski method.
(Hint: You may use the coding of Newton Method given in Moodle pages)
b) Use your coding to find a root of the equation
(?−2)2−ln(?)=0
With initial guess ?0=1.0 and ?0 = 3.0.
Write or print the results in your Homework sheet.

Please prove that: A nonempty compact set S of real numbers has a largest element (called the maximum) and a smallest element (called the minimum).

By the way, I think a minimum is provided by -max(-S)

Theorem 2.1. Cauchy’s Theorem: Abelian Case: Let G be a finite abelian group and p be a prime such that p divides the order of G then G has an element of order p.

Problem 2.1. Prove this theorem.

Question 1

Consider the following two Diophantine equations with integer variables X and Y

(1) 99 X + 225 Y = 36

(2) 225 X + 99 Y = 33

(a) Determine which of these two equations is inconsistent, and explain why
(you can use the Maple commands ifactor and/or gcd ).
For the equation that is consistent, apply Extended Euclid's Algorithm to find a solution  {X0, Y0}
.

(b) Find now all solutions  {X, Y} of that equation, and identify the solution with smallest possible absolute value of  Y
.  

QUESTION 2

Consider the ring of congruence classes R=ℤ/15ℤ:

(a) Identify the set S of all zero divisors and the set U of all units in R. Explain your solution.

(b) For the element [10], find all complementary zero divisors or all inverses, whichever exists. Explain.

(c) Find all solutions of the equation [23] X = [12]
if it is consistent, or explain why this equation is inconsistent.

We consider the operation of the symmetric group S4 on the set R[x,y,z,a] through permutation of an unknown integer.
a) Calculate the length of the orbit of polynomial x2+y2+z+a. How many permutations leave this polynomial unchanged?
b) Is a polynomial of length 5 under this operation possible?
c) Show the existence of polynomials with orbit length 12 and 4.

Brady is a figure skater. He finds a few of the jumps he does to be difficult, but the rest are easy for him. He must include four jumps in his routine. He always makes the first jump a difficult one. After a difficult jump, there is a 0.4 probability that he'll do another difficult jump, and otherwise he'll do an easy one. After an easy jump, there is a 0.2 probability that he'll do another easy one, and otherwise he'll do a hard one.  

What is the probability that the final (fourth) jump of his routine will be an easy one?    

What is the probability that all four jumps of his routine will be difficult ones?    
Enter your answers as whole numbers or decimals.

Let X = {1, 2, 3, 4}, Y = {a, b, c}.

(1) Give an example for f : X → Y so that ∀y ∈ Y, ∃x ∈ X, f(x) = y. 1 2

(2) Give an example for f : X → Y so that ∃y ∈ Y, ∀x ∈ X, f(x) = y.

(3) Give an example for f : X → Y and g : Y → X so that f ◦ g = IY

Let Z₂ [x] be the ring of all polynomials with coefficients in Z₂. List the elements of the field Z₂ [x]/〈x²+x+1〉, and make an addition and multiplication table for the field. For simplicity, denote the coset f(x)+〈x²+x+1〉 by (f(x)) ̅.

Find a particular solution to the differential equation: y'' - 1y' - 20y = -400t^3