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
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 12E: Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and...
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