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Let S = {(x,y) : x, y € R} = R². Define an equivalence relation on S (i.e. a subset of S × S = R² × R?) by (x, y) S (x2, Y2) iff x < x2 and y < y2. (a) Find all elements (x, y) E R² such that (2, T) S (x,y). You might consider drawing a picture in S = R? to illustrate. (Unfortunately, it's difficult to draw pictures of the relation in S × S like in other examples). (b) Find two specific elements (x, y), (x2, Y2) that are not comparable, i.e. (x, y) { (x2, Y2) and (x2, Y2) Z (x, y) (c) Show that S is a reflexive and transitive relation, but not symmetric.

Let S = {(x,y) : x, y € R} = R². Define an equivalence relation on S (i.e. a subset of S × S = R² × R?) by (x, y) S (x2, Y2) iff x < x2 and y < y2. (a) Find all elements (x, y) E R² such that (2, T) S (x,y). You might consider drawing a picture in S = R? to illustrate. (Unfortunately, it's difficult to draw pictures of the relation in S × S like in other examples). (b) Find two specific elements (x, y), (x2, Y2) that are not comparable, i.e. (x, y) { (x2, Y2) and (x2, Y2) Z (x, y) (c) Show that S is a reflexive and transitive relation, but not symmetric.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.4: Ordered Integral Domains
Problem 1E: Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary...
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Let S = {(x,y): x,y E R} = R². Define an equivalence relation S on S (i.e. a subset of S × S = R? × R²) by (x, y) S (x2, Y2) iff x < x2 and y < y2. (a) Find all elements (x, y) E R² such that (2, 7) S (x, y). You might consider drawing a picture in S = relation in S x S like in other examples). R? to illustrate. (Unfortunately, it's difficult to draw pictures of the (b) Find two specific elements (x, y), (x2, Y2) that are not comparable, i.e. (x, y) Z (x2, Y2) and (x2, Y2) Z (x, y) (c) Show that S is a reflexive and transitive relation, but not symmetric.
Transcribed Image Text:Let S = {(x,y): x,y E R} = R². Define an equivalence relation S on S (i.e. a subset of S × S = R? × R²) by (x, y) S (x2, Y2) iff x < x2 and y < y2. (a) Find all elements (x, y) E R² such that (2, 7) S (x, y). You might consider drawing a picture in S = relation in S x S like in other examples). R? to illustrate. (Unfortunately, it's difficult to draw pictures of the (b) Find two specific elements (x, y), (x2, Y2) that are not comparable, i.e. (x, y) Z (x2, Y2) and (x2, Y2) Z (x, y) (c) Show that S is a reflexive and transitive relation, but not symmetric.
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