Let A be a nonempty set with partial order R. For each a E A, let (a) = {x E A| x Ra}. Note: If R was an equivalence relation, then we can think of (a) asthe "equivalence class" of a.1. Show that for a, b e A, a Rb if and only if (a) C (b).2. What can be concluded about a, b e A if (a) = (b)? Explain.3. Let BC Awith B nonempty, and suppose c E A is an upper bound of B. Is (c) an upper bound of F = {(b) | be B}? Note that for this question, theposet we are now using is P(A) with C.

Answered: Image /qna-images/answer/7872c8b8-4e90-4ff2-bc06-84924102d26a.jpg

Answered: e "equivalence class" of a. Show that…

e "equivalence class" of a. Show that for a, b E A, a Rbif and only if (a) C (b). What can be concluded about a, b E A if (a) = (b)? Explain.

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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Let A be a nonempty set with partial order R. For each a E A, let (a) = {x E A| x Ra}. Note: If R was an equivalence relation, then we can think of (a) as
the "equivalence class" of a.
1. Show that for a, b e A, a Rb if and only if (a) C (b).
2. What can be concluded about a, b e A if (a) = (b)? Explain.
3. Let BC Awith B nonempty, and suppose c E A is an upper bound of B. Is (c) an upper bound of F = {(b) | be B}? Note that for this question, the
poset we are now using is P(A) with C.

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