Let A be a nonempty subset of a bounded set B. Why does inf A and supsup A exist? Show that (a) inf B ≤ inf Aand (b) sup A ≤ sup B.

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Answered: Let A be a nonempty subset of a bounded…

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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13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.
Label each of the following statements as either true or false. Every upper bound of a nonempty set is a least upper bound.
Label each of the following statements as either true or false. If a nonempty set contains an upper bound, then a least upper bound must exist in .
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16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.
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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.
Let (A) be the power set of the nonempty set A, and let C denote a fixed subset of A. Define R on (A) by xRy if and only if xC=yC. Prove that R is an equivalence relation on (A).
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Let A be a nonempty subset of a bounded set B. Why does inf A and sup A exist? Show that (a) inf B ≤ inf A and (b) sup A ≤ sup B.