10:03 AM Sat May 21Cancel●●●RESETLet I be the set of real numbers equipped with the standard topology. Let A andB be two subsets of R defined respectively byA = {1,1,2,4}, B = (0, 1) U (1,2).Discuss each of the following questions with justification.(a) Is A open in R?(b) Is B closed in R?(c) Is AUB closed in R?98%Done||||||||||| | | | | | | | | | ||

Answered: Image /qna-images/answer/0cb57c51-3cf3-42ff-b7e4-3014f0019527.jpg

Answered: Let I be the set of real numbers…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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LetO = (1,-2,3,-4,5} and J = {-1,2,-3,4,-5}, define relations N and M from set O to set J as follows, for all (x,y)=0xJ: a. (x,y)e N means that xy is a positive integer 26 169 13 28 2 V121-15 5 b. M =- -13 一年一2
This question refers to unions and intersections of relations. Since relations are subsets of Cartesian products, their unions and intersections can be calculated as for any subsets. Given two relations R and S from a set A to a set B, RUS = {(x, y) E AXB I (x, y) ER or (x, y) E S} RNS = {(x, y)E AXB| (x, y) ER and (x, y) Es}. Let A = {-2, 2, 3, 5} and B = {2, 3} and define relations R and S from A to B as follows: For every (x, y) E Ax B, x R y - |x| = lyl and xsyex - y is even. Using set-roster notation, state explicitly which ordered pairs are in A x B, R, S, RU S, and RN s. (Enter your answers as comma-separated lists of ordered pairs.) AXB = R = S = RUS = ROS =
(a) Give a direct proof of the following statement: "Let A and B be sets. If ACB, then AnB = A."
Let X, Y and Z be subsets of the universal set U. Prove using ONLY definitions (logical connectives) that X \ (Y \ Z) = (XN Z) U (X \ Y). COMPLETE SOLUTIONS TYPEWRITTEN OR WRITE LEGIBLY UNCLEAR SOLUTIONS WILL GET A LOW RATING
For all relations R and S, Ro S = So R. O True O False
Let Q and R be subsets of Q. Prove that the following statements are equivalent: Use element argument to prove (i)=(ii) and (iii) (i). Provide an algebraic proof for (ii)=(iii). (write all the identities and definition) (i) Q is a subset of R. (ii) Q and R are disjoint sets. (iii) The union of Q and R is Q.
Let Q and R denote sets in a universe U. Use algebra of sets to prove the following: an (R - Q) = + QU(R - Q) = Q UR

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Let I be the set of real numbers equipped with the standard topology. Let A and B be two subsets of R defined respectively by A={1,1,2,4}, B = (0, 1) U (1,2). Discuss each of the following questions with justification. (a) Is A open in R? (b) Is B closed in R? (c) Is AUB closed in R?