Exercise 3.5. Let A 3,8), and B 2, 6, C (With of course Ω-R) Find the...
Exercise 3.8. Let A { 1, 3, 5, 7, 9), B-{0.2, 4, 6, 8], C-(1, 2, 4, 5, 7, 8}, and Ω={0.1.2, 3,4,5, 6, 7, 8, 9). Find the following: a An BuC, and An (BUC) (Note that these aren't the same!) b (An B) U (AnC), and An(BUC) c AUB, and AuB. Are these the same?
(2) Find their intersection and union: Let A={1,3,5,7,9}, B = {2, 4, 6, 8, 10}, C = {1,2,3,4,5}, (2a) (a) AnB =? (2b (b) AnC =? (c) AUB=? (2c). (d) AUC =? (20) (4) Express the following compound inequality: * <4 AND 3-1 (a) in set-builder notation:) (4a) (b) (graph it on the number line:) - -3 3 (c) (in interval notation:) (4c) (3) Answer the following questions: (if A and B are two sets) (a) (The keyword AND A...
[1] Let A-11 j Let ถ be the triangle in R, with vertices (-3,-2), (2,3), (-1,1). (a) Find the area of Ω. (b) What is the shape of the image A(S2)? Find the area of A(S) c) Is the linear transformation A orientation preserving or reversing?
[1] Let A-11 j Let ถ be the triangle in R, with vertices (-3,-2), (2,3), (-1,1). (a) Find the area of Ω. (b) What is the shape of the image A(S2)? Find the area...
#s 2, 3, 6
2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
Please help me prove 2,4, and 5. Thank you
Theorem 17. Let A, B and C be sets. Then the following statements are true: (1) AB CA; (2) B CAUB; (3) A CAUB; (4) AB=BA; (5) AU (AUC) = (AUB) UC; (6) An(BNC) = (ANB) nC; (7) An (BUC) = (ANB) U (ANC); (8) AU (BAC) = (AUB) n(AUC).
8. Let P(A) P(B) - 1/3 and P(AnB) 1/10. Find the following: (b) P(AUB') () P(Bn A (d) PA UB)
(a) Let Ω = [4, 101 and let A = 16, 6], [8, 10]} 2. (i) Find F(A) (ii) Let X : 2->R be defined by X = 2-1[4,5]-3 . 1 (6,8) Is X, F(A)-measurable? Justify your answer. (b) Let (2, F) be a measurable space, and let X :2-R. Suppose that X+ is F-measurable. Does this imply that X is F-measurable? Either prove it or give a counterexample.
(a) Let Ω = [4, 101 and let A = 16,...
Exercise 1.1. You ask a friend to choose an integer N between 0 and 9. Let A = {N 〈 5), B = {3 N 7) and C = {N is even and 〉 0). List the points that belong to the following events: (a) AnBnc (b) Au (Bnce) (c) (AUB)nc (d) (AnB)n ((AUC)c)
Finish the proof of Theorem 3.14.
Theorem 3.14 Let (neN aand EneN be sequences in R. Let be in R# and suppose that x" → x, y, → oo, and z" →-oo. . If -oo <x o, then +yn 2. If-oo x < 00, then x" + Zn →-00 4. If-oo x < 0, then xoY" →-00 and xnZn → oo. 5. If x is in R. then-→0and-" →0 Proof Note that the conditions in the different parts of the...
Question 6 (2 pts). [Exercise 4.1.9] Let V = W = R 2 . Choose
the basis B = {x1, x2} of V , where x1 = (2, 3), x2 = (4, −5) and
choose the basis D = {y1, y2} of W, where y1 = (1, 1), y2 = (−3,
4). Find the matrix of the identity linear mapping I : V → W with
respect to these bases.
QUESTION 6 (2 pts). Exercise 4.1.9 Let V = W...