Example 4.13. Consider a set with three elements, s = {a,b, c}. Set A = {(a,a), (b, b), (c, c)}. Essentially, up to permuting the names of the elements, there are 5 partial orders R on S: (1) The elements are linearly ordered: R = Au {(a,b), (a, c), (b, c)}. (2) One element is greater than the two other unrelated elements: R= Au {(b, a), (c, a)}. (3) One element is smaller than the two other unrelated elements: R = Au {(a, b), (a, c)}. (4) Two elements are related, while the third element is unrelated: R=Δυ (α, ). . (5) The elements are not related: R=A. Exercise 4.14. Display each of the partial orders in the three element ex- ample graphically. Decide whether there are minimal or maximal elements, and if so, name them. Decide whether the partial orders have minima or maxima, and if so, name them. Exercise 4.15. Repeat Example 4.13 and Exercise 4.14 with a set of four elements.

Example 4.13. Consider a set with three elements, s = {a,b, c}. Set A = {(a,a), (b, b), (c, c)}. Essentially, up to permuting the names of the elements, there are 5 partial orders R on S: (1) The elements are linearly ordered: R = Au {(a,b), (a, c), (b, c)}. (2) One element is greater than the two other unrelated elements: R= Au {(b, a), (c, a)}. (3) One element is smaller than the two other unrelated elements: R = Au {(a, b), (a, c)}. (4) Two elements are related, while the third element is unrelated: R=Δυ (α, ). . (5) The elements are not related: R=A. Exercise 4.14. Display each of the partial orders in the three element ex- ample graphically. Decide whether there are minimal or maximal elements, and if so, name them. Decide whether the partial orders have minima or maxima, and if so, name them. Exercise 4.15. Repeat Example 4.13 and Exercise 4.14 with a set of four elements.

Name: Please do Exercise 4.15. I have included 4.13 and 4.14 for your refere
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Description: Please do Exercise 4.15. I have included 4.13 and 4.14 for your reference as well. And please keep it short and simple with typefont or legible handwriting, and clear explanations for each step. Thank you! Example 4.13. Consider a set with three elem

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 32EQ

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