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Exercise 1.3.2. Here is a paraphrase of the exercise. Suppose that and y are real numbers. Show that and Remarks max{x,y} = min{x,y} = x + y + x - y 2 x + y − |x − y\¸ - 2

Exercise 1.3.2. Here is a paraphrase of the exercise. Suppose that and y are real numbers. Show that and Remarks max{x,y} = min{x,y} = x + y + x - y 2 x + y − |x − y\¸ - 2

Elementary Geometry for College Students
Elementary Geometry for College Students
6th Edition
ISBN:9781285195698
Author:Daniel C. Alexander, Geralyn M. Koeberlein
Publisher:Daniel C. Alexander, Geralyn M. Koeberlein
Chapter6: Circles
Section6.CT: Test
Problem 11CT: aIf HP=4, PJ=5, and PM=2, find LP. _ bIf HP=x+1, PJ=x1, LP=8, and PM=3, find x. _
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Please solve with explanation... use the remarks below as well, if needed

Exercise 1.3.2. Here is a paraphrase of the exercise. Suppose that and y are real numbers. Show that and max{x,y} = min{x,y} = x + y + x - y 2 x + y − |x − y\¸ - 2 Remarks If Exercise 1.2.10 had max and min in place of sup and inf, then the solution would be easy. The main difficulty in the exercise is that the suprema and the infima may not belong to the respective sets. Nonetheless, there must be elements in the sets within an arbitrarily small positive & of the suprema and the infima. The key to the proof is that establishing equality up to an arbitrarily small error is good enough. Exercise 1.3.2 is a standard application of the method of proof by cases. The trichotomy property of Definition 1.1.1 says that either a <y, or x = y, or > y. The upshot is that the "lattice" operations of maximum and minimum can be obtained from the operations of addition, multiplication, and absolute value.
Transcribed Image Text:Exercise 1.3.2. Here is a paraphrase of the exercise. Suppose that and y are real numbers. Show that and max{x,y} = min{x,y} = x + y + x - y 2 x + y − |x − y\¸ - 2 Remarks If Exercise 1.2.10 had max and min in place of sup and inf, then the solution would be easy. The main difficulty in the exercise is that the suprema and the infima may not belong to the respective sets. Nonetheless, there must be elements in the sets within an arbitrarily small positive & of the suprema and the infima. The key to the proof is that establishing equality up to an arbitrarily small error is good enough. Exercise 1.3.2 is a standard application of the method of proof by cases. The trichotomy property of Definition 1.1.1 says that either a <y, or x = y, or > y. The upshot is that the "lattice" operations of maximum and minimum can be obtained from the operations of addition, multiplication, and absolute value.
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