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In each of the following cases of a set X and relation determine whether ~ is reflexive, symmetric, transitive, an equivalence relation. Give reasons for your answers. i. X = Z, for all x, y ≤ Z, x ~ y ⇒ |x − y| = 3; V m2 m2

In each of the following cases of a set X and relation determine whether ~ is reflexive, symmetric, transitive, an equivalence relation. Give reasons for your answers. i. X = Z, for all x, y ≤ Z, x ~ y ⇒ |x − y| = 3; V m2 m2

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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In each of the following cases of a set X and relation~ determine whether ~ is reflexive, symmetric, transitive, an equivalence relation. Give reasons for your answers. i. X = Z, for all x, y ≤ Z, x~ y ⇒ |x − y| = 3; - ii. X = R², for all (x₁, y₁) and (x2, Y₂) in R² we have (x₁, y₁) ~ (x2, Y₂) if and only if there exists a r > 0 such that (x1, Y₁) = r(x2, Y2). If (ii) is an equivalence relation, then sketch the equivalence class of the point (1, 1) and describe the other equivalence classes.
Transcribed Image Text:In each of the following cases of a set X and relation~ determine whether ~ is reflexive, symmetric, transitive, an equivalence relation. Give reasons for your answers. i. X = Z, for all x, y ≤ Z, x~ y ⇒ |x − y| = 3; - ii. X = R², for all (x₁, y₁) and (x2, Y₂) in R² we have (x₁, y₁) ~ (x2, Y₂) if and only if there exists a r > 0 such that (x1, Y₁) = r(x2, Y2). If (ii) is an equivalence relation, then sketch the equivalence class of the point (1, 1) and describe the other equivalence classes.
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