5. Let f: X → Y. Prove that for any indexed family (Ai);el of subsets of Y iEI iEI iEI iEI 5. Let f: X → Y. Pr...
Let X be a set and let T be the family of subsets U of X such that X\U (the complement of U) is at most countable, together with the empty set. a) Prove that T is a topology for X. b) Describe the convergent sequences in X with respect to this topology. Prove that if X is uncountable, then there is a subset S of X whose closure contains points that are not limits of the sequences in S....
(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;).
(1) Let Ω be a set, and let Ao be a family of subsets of $2. Prove that there exists a minimal-algebra in Ω containing 4). In other wo)rds. prove that there exists a 8 σ-algebra A in 12 such that A C A, and . if A, is any σ-algebra in Ω with Ao c A,, then A c A, (1) Let Ω be a set, and let Ao be a family of subsets of $2. Prove that there...
5. Let A and B be compact subsets of R. (a) Prove that AnB is compact (b) Prove that AUB is compact. (c) Find an infinite family An of compact sets for which UAn is not compact. o-f (d) Suppose that An is a compact set for n 21. Prove that An is compact.
Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1 Let f (x) be a monic polynomial of degree n with distinct zeros ai,..., an. Prove -1
Let X be a finite set and F a family of subsets of X such that every element of X appears in at least one subset in F. We say that a subset C of F is a set cover for X if X =U SEC S (that is, the union of the sets in C is X). The cardinality of a set cover C is the number of elements in C. (Note that an element of C is a...
4. Consider a function f : X → Y. 4a) (5 pts) Let C, D be subsets of Y. Prove that f (CND)sf1(C)nf-1(D). 4b) (10 pts) Let A, B be subsets of X and assume the function f be one-to- one. Prove that f(A) n f(B)Cf(An B) (Justify each of your steps.) 4c) (4pts) Find an example showing that if the function f is not one-to-on the inequality (1) is violated.
(e) Letf: R2-R2 be given by f(a,y) = (V-y,y) Let A, B be the subsets of R2 as indicated in the picture below. Prove that f maps A onto B. (0,1) (1;1) (-1,1) (0,1) v=1 1/2 y-axis y=x2 v-axis v -u b ets t) ide ods.a notteog (0,0) X-axis u-axis (0,0) (e) Letf: R2-R2 be given by f(a,y) = (V-y,y) Let A, B be the subsets of R2 as indicated in the picture below. Prove that f maps A onto...
5-4. Prove that the MGF of Gamma distribution is В f(t) В — t. 5-5. Let X1, X2,... ,X be independent with X~ Gamma (ai,ß). Let Y = EX4. Prove that Y Gamma (Eai,ß)
5. Let X be a topological space and let A and B be connected subsets of X. Prove that if AndB+, then AUB is connected.