2. Let’s consider how the operations (∧) and (∨) relate to the quantifier (∃).

a) Give and explain an explicit subset S of real numbers and
predicates P(x) and Q(x) for which the statement below is false:43
∃x ∈ S, P(x) ∧ Q(x) ↔ [∃x ∈ S, P(x)] ∧ [∃x ∈ S, Q(x)].

Since {1, 2, . . . , 6} is the set of all possible outcomes of a throw with a regular die, the set of all possible outcomes of a throw with two dice is Throws := {1, 2, . . . , 6} × {1, 2, . . . , 6}. We define eleven subsets P2, P3, . . . , P12 of Throws as follows: Pk := {<m, n>: m + n = k} for k ∈ {2, 3, . . . , 12}. For example, P3 is the set of all outcomes for which the sum of the two numbers of dots thrown is 3.

(a) Show that the sets P2, P3, . . . , P12 form a partition of the set Throws.

(b) Let R be the equivalence relation on Throws that has P2, P3, . . . , P12 as its equivalence classes. Give a definition of R by means of a description.

(c) Give a complete system of representatives for the equivalence relation R.

a) Use truth tables to show that the following are valid arguments:

i. [p  (p → q)] → q

ii. [(p → q) ∧ (q → r)] → (p → r)

b) Use truth tables to show the logical equivalence of:

i. (p → q) ⇔ (￢p ∨ q )

ii. (￢p ∨ q) ∨ (￢p  q) ⇔ p

find the particular solution for:

y'' - 4y' - 12y = e^2x(-3x² + 4x + 5) (hint: try y= ue^2x then use: u = Ax² + Bx + C)

Lottery Winnings: State sponsored lotteries are an extremely popular and highly successful method by which state governments raise the much needed funds for financing public expenses, especially education. Needless to say, they are also a very colorful part of everybody’s hopes of striking it rich. States often team up so that the member lotteries can offer higher jackpots to participants. Mega Millions is one of these games, where 44 lotteries team up to offer prizes of at least $12 million. Jackpots are rolled over and grow until someone wins. Mega Millions paid the record jackpot of US lotteries in March 2012, with a jackpot of $656 million, to three winning tickets from Kansas, Illinois and Maryland. The lottery carries two payment options to the winner. Winners can either opt to take 26 equal annual installments, or take the cash payout option at their share of $474 million. There is a 25% federal tax on lottery winnings and a 5% state tax for Kansas and Illinois and 8.75% state tax for Maryland on lottery winnings.-How much would the after-tax annual payment be for each winner?-Each one of these winners chose the cash pay-out option. Assuming a return of 5% a year, did they make the correct decision?-Is the lottery correct in advertising the jackpot at $656 million?-If the lottery would like to give the annuity option a chance at being selected, how do you think they should structure their payment plans? Any ideas?

In this problem we consider another way to think about the rational numbers. Normally we would write fractions as p/q for p ∈ Z and q ∈ N. In this problem we represent fractions as ordered pairs. So let S = {(p, q)|p ∈ Z and q ∈ N}.

For ordered pairs (p, q) and (r, s) in S define (p, q)R(r, s) if and only if ps = qr.

You should think about how this is related to the test that two fractions are equal.

a. Prove that R is an equivalence relation on S.

b. What is the equivalence class that contains (0, 1)?

c. What is the equivalence class that contains (2, 1)? Now define a partial order (p, q) ≤ (r, s) for (p, q) ∈ S and (r, s) ∈ S. Answer each of the following question and prove your result.

d. Is this a reflexive relation?

e. Is it symmetric?

f. Is it antisymmetric?

g. Is this a transitive relation?

Now assume that the harvesting is not done at a constant rate, but rather at rates that vary at different times of the year. This can be modeled by ??/?? = .25? (1 − ?/4 ) − ?(1 + sin(?)). This equation cannot be solved by any technique we have learned. In fact, it cannot be solved analytically, but it can still be analyzed graphically.

8.) Let c=0.16. Use MATLAB to graph a slopefield and approximate solutions for several different values of p(0), and interpret what you see. Turn in the graphs together with your analysis. Note: 0<p<5, and 0<t<50 is a reasonable viewing window to start with. PLEASE DO IT USING MATLAB.

State and prove spectral mapping theorem

At the end of every year an investor pays £2,000 towards additional voluntary contributions to build up a private pension fund. The investor intends to retire in 30 years and wants the pension fund to contain at least £100,000 at the date of retirement. What is the annual effective rate at which the contributions should accumulate? (Perform few steps of both the bisection method and the interpolation method with suitable starting values)

Is the given set of vectors a vector subspace (Give reasons)? If your answer is yes, determine the dimension and find a basis. All vectors in R5 with v1 + 3v2 - v3 = 0, 3v1 + v2 - v4 = 0, 4v1 + 2v2 - v5 = 0 (v1, v2, … denote components). Show details.

Answer the following parts in Plane Geometry: (a) Show that the converse to the alternate interior angles theorem postulate implies the angle sum postulate in Plane Geometry. (Hint: Check out Euclid Book 1 Prop 32 for the idea, but make sure you write an axiomatic Plane Geometry argument.) (b) Show that the angle sum postulate implies the converse to the alternate interior angles theorem. (Hint: Draw a perpendicular.) (c) Explain why that means you can conclude that the two statements are equivalent to each other in Plane Geometry.

Let Xn be the Markov chain with states S = {1, 2, 3, 4} and transition matrix.

a.) Let X0 = 3 and let T3 be the first time that the Markov chain returns 3, compute P(T3 = 2 given X0=3). Please show all work and all steps.

b.) Find the stationary distribution π. Please show all work and all steps.

for the system equation of x' = Ax   if Coefficients Matrix A be ? = [ 5 −5 −5 −1 4 2 3 −5 −3 ] , find the basic matrix

a) Solve the Cauchy-Euler equation: x^2y'' - xy' + y = x^3

b) Solve the initial-value problem: y'' + y = sec^3(x); y(0) = 1, y'(0) =1/2