Problem 4 Let A be a set, and let R be a relation on A. 1. Suppose R is reflexive. Prove that ∪x∈A[x] = A. 2. Suppose R is symmetric. Prove that x ∈[y] if and only if y ∈[x], for all x, y ∈A. 3. Suppose R is transitive. Prove that if xRy, then [y] ⊆[x] for all x, y ∈A.

Problem 4 Let A be a set, and let R be a relation on A. 1. Suppose R is reflexive. Prove that ∪x∈A[x] = A. 2. Suppose R is symmetric. Prove that x ∈[y] if and only if y ∈[x], for all x, y ∈A. 3. Suppose R is transitive. Prove that if xRy, then [y] ⊆[x] for all x, y ∈A.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 6TFE: Label each of the following statements as either true or false. Let R be a relation on a nonempty...

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Question

Problem 4
Let A be a set, and let R be a relation on A.
1. Suppose R is reflexive. Prove that ∪x∈A[x] = A.
2. Suppose R is symmetric. Prove that x ∈[y] if and only if y ∈[x], for all x, y ∈A.
3. Suppose R is transitive. Prove that if xRy, then [y] ⊆[x] for all x, y ∈A.

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