Please solve Exercise 2.1.7... parts a and b with detailed explanations Exercise 1.4.6a. Show that every closed interval [a, b] is the intersection of countably many openintervals.b. Show that every open interval (a, b) is a countable union of closed intervals.c. Show that an intersection of a possibly infinite family of bounded closed intervals,[ax, bx], is either empty, a single point, or a bounded closed interval.λειExercise 2.1.7a. Show that lim n = 0 (that is, the limit exists and is zero) if and only ifn→∞olim || = 0.n∞0b. Find an example such that {n} converges and {n}₁1 diverges.Remarks1In parts (a) and (b) of Exercise 1.4.6, the intervals can be chosen to be nested.In part (c) of Exercise 1.4.6, the conclusion is an "or" statement (in fancy language, adisjunction). Parsing the logical structure reduces the problem to proving that if theintersection contains at least two points, then the intersection must be a bounded closedinterval.The point of part (b) of Exercise 2.1.7 is that the value 0 is special in part (a).

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Answered: Tax, ox], is either empty, a single…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 23E

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21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.
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Tax, ox], is either empty, a single point, or a bounded closed interval. ΧΕΙ Exercise 2.1.7 a. Show that lim n = 0 (that is, the limit exists and is zero) if and only if n→∞0 lim n = 0. b. Find an example such that {n} converges and {n} diverges. Remarks