Consider the relation T on the set of all undergraduate TAs (UTA) for the CSE department where s₁ T s₂ if and only if s₁ and s₂ are UTAs for the same course.

(a) Assuming that no one is a UTA for multiple courses, prove that T is an equivalence relation.

(b) Assuming that no one is a UTA for multiple courses, what do the equivalence classes for T represent?

(c) Explain why the assumption that no one is a UTA for multiple courses is necessary for T to be an equivalence relation.

1. (a) Explain what is meant by the terms countable set and uncountable set. Give a concrete example of each.

b) Show that if A and B are countably infinite then the set A x B is also countably infinite.

c) Give an expression for the cardinality of set A U B when A and B are both finite sets.

d) What can you say about the cardinality of the set A U B when A and B are infinite sets.

Determine whether the polynomial f(x) is irreducible over the indicated field. (a) f(x) = 4x^ 3 + 10x^2 + 16x − 11 over Q (b) f(x) = 7x^ 4 − 9x^ 2 + 6 over Q (c) f(x) = x^3 − x + 1 over Q (d) f(x) = x ^4 + 2x ^2 + 2 over Q (e) f(x) = x^ 4 + 2x^ 2 + 2 over C (f) f(x) = x ^4 + 2x^ 2 + 2 over R (g) f(x) = x ^2 + 2x − 3 over Z-5 (h) f(x) = x ^3 − 2 over Z-7

Polynomial and rational functions can be used to model a wide variety of phenomena of science, technology, and everyday life.

Choose one of these sectors and give an example of a polynomial or rational function modeling a situation in that sector. [Hint: see the examples and exercises in the book.]

Go to www.desmos.com/calculator, write your equation, or function, and develop your explanation using the properties of graphs.

Your Discussion should be a minimum of 250 words in length and not more than 750 words.

Dear expert, I will submit this assignment after an hour, please don't copy from internet and add APA text citation if needed.

Let F be a field and let φ : F → F be a ring isomorphism. Define Fix φ to be Fix φ = {a ∈ F | φ(a) = a}. That is, Fix φ is the set of all elements of F that are fixed under φ. Prove that Fix φ is a field.   (b) Define φ : C → C by φ(a + bi) = a − bi. Take for granted that φ is a ring isomorphism (we proved this in class at some point). Find Fix φ.

Use the method of reduction of order to find a second solution y2 of the given differential equation such that {y1, y2} is a fundamental set of solutions on the given interval.

t²y′′ +2ty′ −2y=0, t > 0, y1(t)=t

(a) Verify that the two solutions that you have obtained are linearly independent.

(b) Let y(1) = y0, y′(1) = v0. Solve the initial value problem. What is the longest interval on which the initial value problem is certain to have a unique twice differentiable solution?

(c) Now suppose that y(0) = y0, y′(0) = v0. Is there a solution in this case (how does it depend on y0 and v0? If so, how many? Explain.

Let R and S be commutative rings with unity. (a) Let I be an ideal of R and let J be an ideal of S. Prove that I × J = {(a, b) | a ∈ I, b ∈ J} is an ideal of R × S. (b) (Harder!) Let L be any ideal of R × S. Prove that there exists an ideal I of R and an ideal J of S such that L = I × J.

Fill in the missing algebraic steps in the concrete renormalization calculation for period doubling.Let f(x) = -(1 + mu)x+x^2. Expand p+(n)subn+1 =

f^2(p+(n)subnetting) in power of small deviations (n) subn using the fact that p is a fixed point of f^2

Q1. Define/outline the following:

a. the difference between a ‘one-tailed test’ and a ‘two-tailed test’.

b. the importance of sample size in the context of OLS regression.

*c. four different types of data structures and discuss their potential usefulness and application in finance.

d. the OLS assumptions

e. Correlation is not causation. Briefly discuss.

(c. is the part I'm confused most about)

5. Find the generating function for the number of ways to create a bunch of n balloons selected from white, gold, and blue balloons so that the bunch contains at least one white balloon, at least one gold balloon, and at most two blue balloons. How many ways are there to create a bunch of 10 balloons subject to these requirements?

You wish to retire in 18 years , at which time , you want to have accumulated enough money to receive an annuity of $500 monthly for 20 years of retirement. During the period before retirement you can earn 4% annually,while after retirement you can earn 6 percent on your money. What monthly contributions to the retirement fund will allow you to receive the 500 dollars annuity?

X and Y are subsets of a universal set U.

Is the following statement true or false? Support your answer with a venn diagram.

X - (Y u Z) = (X - Y) n (A - C)

Answer each one, with (brief) justification. Throughout, V denotes a vector space, and bold-face letters like u, v, etc denote vectors in V.

A1. Suppose there are three linearly independent vectors in V. Is V necessarily finite-dimensional?

A2. Suppose there are three vectors in V that span V. Is V necessarily finite-dimensional?

A3. What is the dimension of the vector space of all polynomials of degree 4 or less?

A4. Can an infinite-dimensional vector space contain a finite-dimensional subspace?

A5, In a finite-dimensional vector space, is every spanning set necessarily finite?

A6. Can R³ contain a four-dimensional subspace?

Solve the following equations in complex numbers and write your answer in polar and rectangular form.

(a) z² − i = 0 in C

(b) z⁷ + z⁶ + z⁵ + z⁴ + z³ + z² + z = 0

(c) e^2z + 2e^z = −2

(d) z + 1 / z−1 = e^iπ/3

2. Define a relation R on pairs of real numbers as follows: (a, b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements.

3. Define a relation R on integers as follows: mRn iff m + n is even. Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements.

4. Define a relation R as follows:

R = {(a, a),(b, b),(c, c),(d, d),(c, a),(a, d),(c, d),(b, c),(b, d),(b, a)}

Is R a partial order? Why or why not? If R is a partial order, draw a diagram of some of its elements.

5. How many different partial ordering relations are there on the set {a, b, c}?

6. Which of the following relations are partial orderings? Which are total orderings? Which are well-orderings?

            a. The relation described in Problem 2

            b. The relation described in Problem 3

            c. The relation described in Problem 4

            d. The "less-than" relation on the integers

            e. The "less-than-or-equal" relation on the integers

            f. The "less-than-or-equal" relation on the natural numbers

            g. The "less-than-or-equal" relation on the real numbers

            h. The "less-than-or-equal" relation on the non-negative real numbers

            i. The relations in Problem 5