Please find the following attachment.
2. Suppose A,B,C,D are sets. We know that (A x B)U(C x D) S (AUC) x...
Prove or disprove: for all sets A, B, C and D, (Ax B) U (Cx D) (AUC) x (BUD).
Prove equalities involving sets A, B, C and D a) (AIB)U(C1B) = (AUC) IB b) (AUB)-(ANB) = (A-8)U(-A) c) (AxB) OLC xD) - (ANC) x (BND) d) (AXB) (BAA) = (ANB)X(AMB)
Question4 please (1). Let f: Z → Z be given by f(x) = x2. Find F-1(D) where (a) D = {2,4,6,8, 10, 12, 14, 16}. (b) D={-9, -4,0, 16, 25}. (c) D is the set of prime numbers. (d) D = {2k|k Ew} (So D is the set of non-negative integer powers of 2). (2). Suppose that A and B are sets, C is a proper subset of A and F: A + B is a 1-1 function. Show that...
Suppose that 01, ..., In are independent realizations from N (u,02). We know that X ~ N(N ). Suppose we are interested in minimizing the objective function h with respect to a where h is defined as h(a) = (Eſa X – ])? + Var(aĚ) Specifically we are interested in finding the global minimizer â, i.e., a = argmin h(a), CER where R+ denote the set of positive real numbers. (a) (7 points) Find Vh(a) and v2h(a). (b) (7 points)...
Please prove 3. a. Let A = {a,b,c} and B-{b, d). he following six power (parts) sets: P(A), (B). P(AUB). POAB), PCA PCB), and P(A) n (B) b. Let A and B be any two subsets of the same universal set U (not the same sets used in part a.) 1. Using the sets above as an example (or using more examples you can build on your own), make a conjecture about the relation between the sets (A) (B) and...
Let U ={a, b, c, d, e, f, g, h, i, j, k}. Let A={d, f, g, h, i, k}. Let B={a, d, f, g, h}. Let C={a, c, f. i, k} Determine (AUC) U ( AB). Choose the correct answer below and, if necessary, fill in the answer box in your choice. OA. (AUC) U(ANB)= } (Use a comma to separate answers as needed.) OB. (A'UC) U (ANB) is the empty set. LE This Question: 1 pt Let U={x|XEN...
17-26 true or false questions 17. The smallest positive real number is c, where c = card(0,1). 18. To show that two sets A and B are equal, show that x A and x B. 19. If (vx)P(e) is false, then P(x) is never true for that domain. 20. If R is a relation on A and if (a, a) is true for some a in A, then R is reflexive. 21. If f:A → B is a function, then...
Suppose that X - Exp(2), for some a > 0. We know that the moment generating function of X is given by M(O)= E[em]= , for some appropriate set of values of t. (a) Derive this mgf result. Explain what condition ont is necessary for this expression to be valid, and why this condition is necessary. (b) Use the mgf to find the first four moments (u, us, and u) of X. (c) Use your results in part (b) to...
2-t 2. Suppose that X - Exp(2), for some i >0. We know that the moment generating function of X is given by M(t)= E[e"]=-4, for some appropriate set of values of t. (a) Derive this mgf result. Explain what condition ont is necessary for this expression to be valid, and why this condition is necessary. (b) Use the mgf to find the first four moments ( u u , and ) of X. (c) Use your results in part...
Consider the following simple regression model: a. Suppose that OLS assumptions 1 to 4 hold true. We know that homoskedasticity assumption is statedas: Var[UjIx] = σ2 for all i Now, suppose that homoskedasticity does not hold. Mathematically, this is expressed as In other words, the subscript i in σ12 means that the conditional variance of errors for each individual i is different. Under heteroskedasticity, we can derive the expression for the variance of Var(B) as SST Where SSTx is the...