1. (a) Let A be the subset of R defined by A := {1-r:neN}. Find all cluster points of A and justify your answer. (b) Let S cR and let e e R be a cluster point of S. Prove that for each e > 0 the set W := sn(c-E,e +e) has infinitely many elements.
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- 2 Let A be a fixed nonempty subset of set X and I = {B c X | A Z B}. Show that (X,T) is a matroid.(1) Given that A = {x : x is even and less than or equal to 10} andB = {x : x is a multiple 3 and less than or equal to 10} in the universe U={1.,2,3….10}Find A, B, AUB, (AUB)’, A∩B, (A∩B)’, A-B, B-ASuppose that SN = (0, 1] and let B, denote the collection of all sets of the form 2. (a1, b1] U (a2, b2] U..U (a, bu] where k EN is finite and 0 < aj2.28 let xa be a translation of R Then (1) La is a continuous bijection from onto R (ii) The image of an open set under Da is an oper set. of (iii) let & be an open set. The component interval +a are exactly The images of the component intervals of the set & under translation a the goal of this section is to establish the following result.Let ü= Compute ü-ü. 3.2) Let n E N. Suppose An = {n, n+ 1, n +2, ...} and T = {0, An}neN• (a) Show that t is a topology on N, (b) What are closed sets in t? (c) List the open sets containing the positive integer 6. (d) What is the closure of {4} and {2,4,6, 8, . ..}?Let S be a set of 100 points in the plane and the distance between any two points of S is at least 1. Show that these 100 points can be divided into 4 groups such that any two points in the same group has a distance strictly larger than 1.Let S be the subset of the ℝ2 given by all pairs (x, y) so that x2 + y2 = 2. Show that S is a closed subset of ℝ225) Let A be dense subset of (X, T) and let B be non-empty open subset of X thenLet n be a positive integer and let Sn be any set with |Sn|= n. Define Dn to be the digraph with V (Dn)= P (Sn) , the set of all subsets of Sn where (X, Y) element of A(Dn) if and only if X contains Y properly as a subset. a) Make a pictorial representation of D3 b) Prove that D has a unique source. c) Prove that Dn has a unique sink. d) Find a necessary and sufficient condition for Dn to have carrier vertices. e) Find a formula for the size Dn in terms of n. f) Prove that D has no circuit.6. if x C y. (a) (b) Let R = {a: a is a cut}.Let x, y, R. We say that a3 Consider [0, 1] × [0, 1] with dictionary ordering. Find Ã, when A = {(x, y) : 0SEE MORE QUESTIONSRecommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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