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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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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