
ove SПаптеу теw опе теаЅОЛПs 10 Пепп тасе. Exercise 7. Let A and B be sets....
ILULIITUL 10.37 Theorem. (The Generalized Distributive Laws for Sets of Sets.) Let S be a set and let be a non-empty set of sets. Then: (a) SNU =USNA: AE}. (b) Sund= {SUA:AE). Proof (a) Let = {SNA: AE }. We wish to show that S U = UB. For each 1, we have BESUS iff x S and 2 EU iff xe S and there exists AE such that EA iff there exists AE such that reS and x E...
Let f(x) = { 80 -5 if < 10 - 7+ + b if : > 10 If f(x) is a function which is continuous everywhere, then we must have b = Let f(x) = 82 - 5 if x < 10 1 - 7x +b if x > 10 If f(x) is a function which is continuous everywhere, then we must have b= -6 2-5 - 2x + b if - 1 Let f(x) if 2 - 1 There...
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
(5) Let A, B and C be sets. Show that there is a bijection between the sets F(A, B x C) and F(A, B) x F(A, C)
Exercise 4.9. Let X ~ Poisson(10). (a) Find P(X>7). (b) Find P(X < 13 X > 7).
A . Prove that Problem 4. (2 points) Let A and B be two sets. Suppose that A B = B A = B. Problem 5. (optional but recommended). Show that the set X = {(...) 21: sequences of O's and I's is not countably infinite. Hint: think of a natural function between X and P(N). € {0,1}} of infinite
Exercise 1.3. Let {A,:y er be a family of sets indexed by ſ. a) Show that 4, C U4, Wyel. b) Show that n A, CA,, Vyer. Yel mer
Exercise 3.2.12. Let A be an uncountable set and let B be the set of real numbers that divides A into two uncountable sets; that is, s E B if both {{ : 2 € A and r < s} and {x : x € A and x > s} are uncountable. Show B is nonempty and open. T
Exercise 6.B.3. Let the pair of random variables (X, Y) have joint density function f(x, y)-16(x-y)2 įf x, y e [0, 11, 0 otherwise. a. Confirm that f is a joint density function by verifying that equation (6.B.4) holds, and use a computer or graphing calculator to sketch its graph. b. Compute the marginal density function of Y c. For each x e [0,1], compute the conditional density of Y given x. d. Compute the conditional expectation function E(Y|X =...
#1 & #2
Exercise 1. This exercise builds on the method used to prove that if a function differetiable at a point b, then it is also continuous at b. Suppose g : (-1,1) → R is a function such that g(0) = 7 and lim 9)-7-10 exists. Define G())7-10 on-l < x < 1 when x need to know the value of λ, but its existence is necessary in what follows. 0. Let λ be the limit of G(x)...
Exercise 3 (Cantor-Bernstein-Schröder). Let f: A → B and g: B → A be injective maps. We define recursively the sets C = UCn Co = A \ g(B), Cn+1 = g(f(Cn)), nƐN and a new map h: A → B by if x E C, f (x) h(x) = if x 4 C, g='(x) where the preimage g¬1(x) is well-defined since g is injective and x E g(B) in that case (check that!). Show that h is bijective. Conclude...