Prove that if two sets X and Y are equipotent then their power sets P(X) and P(Y ) are equipotent.
Prove that if two sets X and Y are equipotent then their power sets P(X) and P(Y ) are equipotent.
Exercise 4.5. Prove that for every sets X and Y in R"if X & Y then cl(X) sd(Y).
3. Let X and Y be countably infinite sets. (a) Prove: If X and Y are disjoint then XuY is countably infinite. (b) Is the statement in (a) still true if we remove the hypothesis that X and Y are disjoint? If yes, justify your reasoning with a few sentences. If no, provide a counterexample. (P.S. "Counterexample” means that you have to explain why the example you provide demonstrates that the statement is false.)
Let X, Y be two nonempty sets and let f : X → Y. For a, b X we write a ~ b iff f(a) = f(b). Prove that~is an equivalence relation on X Write lely for the equivalence class of x e X with respect to “~" Express [ely in terms of the function f: Irl, = {re x : f(z') a: b: ?? J. (I d o not want to see ..|x ' = {x"e X : r,...
X, Y, Z are three sets in a sample space S. Find P(X|Y ∩Z) if P(X|Y ) = 0.1 and P(X|Z) = 0.35 are given.
Prove that for any two real numbers x and y, |x + y| ≤ |x| + |y|. Hint: Use the previously proven facts that for any real number a, |a|≥ a and |a|≥−a. You should need only two cases.
2. Suppose S and T are nonempty sets of real numbers with the following property: (P) There exists y in T such that for all x in S, y>x. Prove the following (of course, use the standard approach for proving universally and existentially quantified statements): (Q) For all x in S, there exists y in T such that y>x.
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
(1) Let X and Y be sets. Let f be a function from X to Y, (a) IF BEY, recall that F-'(B) = {xeX \flyeBX(y,x) ef-)}. Prove that f'(B)={xeX | fk)e B}. (hint: Reprember that even though t is a thought is a function, the relation f may well not be itself a function.) Al b) Let {B; \je J} be an inbred family of subsets of Y. Prove that of "b) = f'(21B;).
Prove that for all sets M, N, and P, if M UN MU P and MON Mn P, then N P
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =