Let A, A1, A2,... be measurable sets. Then P(A)=1- P(A); • P(Ø) = 0; P(A1 UA2) ≤ P(A1) + P(A2); A1 C A2 P(A1) P(A2); P(UA) + P(n=14) = 1. Exercise 3.1 Prove these relations. ☐
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- d) Give a recursive definition of the sequence {a,}, n = 1, 2, 3... n if a,= 2 + 1 ; a10, and a15 e) Use your definition in (d) above to find f) Let A = {1, 2, {{1,2}}}. Find the power set P(A)B₂ {(₂1 X12 X22 | Xij ≥ 0 for i, j = 1, 2, X11 + 12 = 2, X21 + x22 = 2, X11 + x21 (6) Let B₂ to be the set described in the screenshot attached Suppose Cij are real numbers such that C11C22 C12 + C21 > 0. Consider the following LPP: maximize: C11x11 + C12x12 + C21%21+ C22X22 subject to: (#11 #12) € B₂ X21 = 1.5, x12 + x22 = 2. (a) Write this problem in canonical form {x = (x11, 12, X21, X22) € R4 | Ax = b, x ≥ 0}. Hint: A should be a 3 x 4 matrix (after you eliminate one row which is linearly dependent on the others). (b) List all BFS. Which BFS x* has the maximum cost for this LPP? 2.5}. For the maximizer x*, show that for all basic directions D₁ at x* the reduced cost Cj = c² Dj ≤ 0.Show that if AB={0}, then P(A)≤P(B).
- (4) Let A, B, C, D be sets. (a) (b) Prove that Ax (B \ C) = (A × B) \ (A × C). Prove that (A x B) n (C x D) = (ANC) × (BND).Exercise 3.4.6. Let A be a non-empty family of sets and let B be a set. (1) Prove that (U X∈A X)−B = U X∈A(X −B). (2) Prove that (∩ X∈A X)−B = ∩ X∈A(X −B).•.. 1. Let A = {b,c,d,e} and B = {a, b}. (a) Find A U B. (b) Find A - B. (c) Find P(A) (power set of A). (d) Find A X B. Repl...