R × R | x < y} . This means that R 10. Let R< = {(x, y) relation on R. is the "less than" 95 (a) What is the domain of the relation R<? (b) What is the range of the relation R<? (c) Is the relation R a function from R to R? Explai. Note: Remember that a relation is a set. Consequently, we can talk about one relation being a subset of another relation. Another thing to...
3. Let f : (a,b) +R be a function such that for all x, y € (a, b) and all t € (0.1) we have (tx + (1 - t)y)<tf(x) + (1 - t)f(y). Prove that f is continuous on (a,b).
Looking for full solutions of question (a)(b)and(c) Problem 4. Let A CR. Given any a, b € R, with a <b, define (a,b) A = {2 € A:a < x <b}. We call (a, b) A an open interval in A (or an open interval relative to A). a) Show that (a, b) A = (a,b) n A. b) Prove that a set V C A is relatively open in A if and only if for every 3 EV, there...
3. (15 pts) Let D be an infinite set with cardinal d. Let A = {X C D | 0(X) <3}. Prove that o(A) = d.
Problem 2. Let f be a self-map on a set X. For x,y e X define x ~ y if and only if f"(x) = f(y) for some integers n, m > 0. Show that ~ is an equivalence relation.
3) Define the relation <on R via x < y if and only if xy < 10. Show that is symmetric. (20 points)
3. (a) (5 points) On the set A= R\{0}, let x ~ y if and only if x · y > 0. Is this relation an equivalence relation? Prove your answer. (b) (5 points) Let B = {1, 2, 3, 4, 5} and C = {1,3}. On the set of subsets of B, let D ~ E if and only if DAC = EnC. Is this relation an equivalence relation? Prove your answer.
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
0, P - 0 2 ) - 1. 3. Let (X, A,) be a complete measure space. Assume that A, B E A with (A) = (B) < . Show that if A CCCB, then CE A. 4 Let A and Rhe two collections of euheete of Y Aceume that any cot in 4 halanes
Let X and Y b Var(Y) (1) If a, b,c and d are fixed real numbers, = E(X), μγ E (Y),咳= Var(X) and e ranclom variables. with y a) show Cov(aX +b, cY +d)- ac Cov(X,Y) (b) show Corr(aX + b, cY + d)-PXY for a > 0 and c > 0.