Q1: (a) If (M, d) be a metric space and AC M. Prove that A is the largest open set in Mthat contained in A. then give an example to show that (A U B) # A* U B.(b) let (M, d)be a metric space and let A ≤ M. Then (AB) ≤ĀNB

The interior, or (open) kernel, of A is the set of all interior points of A: the union of all open…

Answered: Q1: (a) If (M, d) be a metric space and…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 18E: Let (A) be the power set of the nonempty set A, and let C denote a fixed subset of A. Define R on...

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Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
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A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
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Q1: (a) If (M, d) be a metric space and A M. Prove that A is the largest open set in M that contained in A. then give an example to show that (AUB)* # A U Bº. (b) let (M, d) be a metric space and let ACM. Then (ANB) ≤ANB