Let U be a set, and let A CU. Recall the indicator function XA: U +...
2. Let U be a set, and let A CU. Recall the indicator function XA: U → Z2 defined by XA() : S 1, XEA 10, x¢ A. Now, let A, B CU and consider the symmetric difference of A and B defined by A A B = (A – B) U (B – A). (a) Show that A AB CU, and compute Ø A A. (b) Prove that Vx € U, XAAB(x) = x1(x) + XB(x), where addition is...
5. Let A, B, C be subsets of a universal set U. Recall for D CU that XD denotes the characteristic function of D. Prove that XAUBUC = XA +XB+XC - XAMB - XAOC · XBNC + XANBNC. Hint: Facts that you may use: (1) Xpe = 1 – Xd. (2) (AU BUC)° = A n Bºn Cº. (3) XEnF = XEXF. (4) XEnFnG XEXFXG. Don't prove these facts.
Let A, B, C be subsets of a universal set U. Recall for D C U that XD denotes the characteristic function of D. Prove that XAUBUC = XA + XB+XC - XACB - XAC - XB C +XAOBOC. Hint: Facts that you may use: (1) XD 1-XD. (2) (AU BUC)° = ACB 1C. (3) XEnF = XEXF. (4) XEnFnG = XEXFXG. Don't prove these facts.
Let frutiv be a continuous, complex-valued function on the connected open set U. Which of the following statements is equivalent to the analyticity of f on U? o af For every zo EU, there are coefficients ao, al, ... such that > an(z – zo)" n=0 converges for every z EU, Uz = Vy and vz = Wy == 0 for all rectangular paths y in U. None of these.
probelms 9.1 9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
Sets A, B, and Care subsets of the universal set U. These sets are defined as follows. U= {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1,6,7,8,9} B = {1, 3, 4, 7, 9) C = {4, 5, 6, 7} Find CU ( BA)'. Write your answer in roster form or as Ø. CU (BNA): = 0 Х 5 ?
Recall that a subspace S of R" has the following subsoace properties. 1. The zero vector 0 is in S 2. If u and v are in S, then u + v is in S. 3 Ifc is a scalar and u is in S, then cu is in S. If a set S of points in Rn doesn't have one or more of these three properties, then S is not a subspace of R Select each statement from the...
Problem 2: (Topological Characterization of Continuity) Let : R → R be a function. Recall that for a subset BCR, we have the set (B) := ER: (a) e B). Prove that is continuous if and only if f'(U) is open for all open sets U CR. Hint: you can use the characterizations of continuity from Theorem 4.3.2 in our textbook
(a)-(d)? Problem(11) (10 points) Let Z~Normal(0, 1). Recall the definition of -value, i.e., P(Z>)-r. (a) (1 point) Find the probability of P(-2a/2<Z < 2a/2) (b) (3 points) Let X1, X2, , Xa be a random sample from some known) mean p and (known) variance o2. Based on the Central Limit Theorm and part (a) above, show that the confidence intervals for the population mean u can be estimated by population with (un- P(x- <pAX+Za/2 =1-a. Za/2 (c) (2 points) The...
13 pts) Let R be the relation on R deÖned by xRy means "sin2 (x) + cos2 (y) = 1". Recall the Pythagorean identity: 8u 2 R we have sin2 (u) + cos2 (u) = 1. (a) (9 pts) PROVE that R is an equivalence relation on R. (b) (4 pts) Describe all elements of the (inÖnite) equivalence class [0]. Recall: sin(0) = 0 and cos(0) = 1. 2. (13 pts) Let R be the relation on R defined by...