Solve these quiz plz

Notes: This homework consists of 5 problems. Write your answers in full sentences. Before you start a problem set up the goal and explain how will you proceed and why. Write all details thinking that the grader knows nothing about Mathematics. Pictures and graphs are great ideas to start a proof but will not be sufficient without written explanation. If you are referring to result or theorem, please cite. 1. Let the universal set be S = {0, 1, 2, 3,4,...} and consider the sets A, B, C and D where A = {1, 4, 7, 10, 13, 16,...}, B= {ne S : n is odd } C = {re S : x is not prime} and D = {1, 2, 3, 5, 8, 13, 21, 34,55,.. . }. Which of the following statements are true? Give a complete explanation for each. (1) 25 6 A (ii) 76 E D (iii) 100 # AUD (W) CCB (v) CAB -0 2. Construct a nontrivial open sentence P(n) over the domain S = {1, 2,5,7} such that P(n) is true for half of the integers in S and false for the other half. 3. Consider the open sentences for which the domain of both m and n is Z: P(m,n) : m' - n' =0, Q(m,n): m =n, and R(m, n): m+n=0. State each of the following statements in words and indicate whether it is true or false. Explain (i) Q(5,5)v R(5,5) (ii) P(3,4) A (~ R(3,4)) (iii) Q(1,1)= P(1,-1) 4. Consider the statement: P: "If John takes Roger to the ball game, then Mary will walk John's dog." For each of the following scenario, determine whether P is true or false. Explain. (a) Either John doesn't take Roger to the ball game or Mary walks John's dog- (b) The ball game is canceled. (c) Mary walks John's dog. (d) Roger does not go to the ball game. 5. For two statements P and Q, construct a truth table for each of the following statements (a) (Q = P) = (~Q) (b) (Q AP) = (Q = P) (c) (~Q) V (P = Q) A (Q = P)1. (4 points) State whether the following statements are true or false. It they are true, briefly explain why. If they are false, either give an example showing the incorrectness of the statement, or briefly explain why. (a) sin(3x) = 3sin(x) (b) The point on the unit circle whose coordinates are (cos(-200"), sin(-200") ) is in the second quadrant. (c) If # is in the interval [0, 27) and cos(8) = 12, then # must be 5. (d) The midline of a periodic function is the horizontal line y = merimen +minimum 2 (9.4) (6 points) State the amplitude, midline and period of the following graph, and write the formula. 2 5. (6 points) State the vertical asymptotes and period of the following graph, and write the formula. -2For observations {Yi, X.},, recall that for the model Yi = do + Boxitei, (1) the OLS estimator for {oo, Bo}, the minimizer of EL, (Y, - Q - BX;), is B_ Ci ( Xi - X) ( Y; - Y) and a = Y - BX. (2) Et (X - X) When the equation (1) is the true data generating process, {X/}7 , are non-stochastic, and {ez, are random variables with E (e;) = 0, E (e;) = 07, and E(ege;) = 0 for any i, j = 1, 2, ..., n and i f j, we know that E (B) = Bo, E(a) =00, Var (B EL, (X, - X) (3) Var (a) n + EL (X - X) and Cov (a, B) = EM, (X - X)' gg can be unbiasedly estimated by 2 [(X -a - BX.)' (4) and Var (B), Var (a), and Cov (a, B), the estimates for Var (3), Var (@), and Cov (a, B), can be computed by replacing of with a in equation (3). 1. For observations {Yi, Xi}T_1, consider another model Yi = on + ei. (5) (a) To estimate on, what is your 'least squares' criterion function? And what is your 'least squares estimator? (b) Suppose that the equation (5) is the true data generating process, {X}, are non-stochastic, and (e}" are random variables with E (e;) = 0, E (e?) =07, and E (eje;) = 0 for any i, j = 1, 2, ..., n and i / j. What is the expected value of your estimator? What is the variance? (c) Following b., let e, denote the residuals. What is the expected value of e,? What is the variance? (d) Following b. and c., suppose that two different estimators for of have been proposed