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1. Let A be a set and suppose R is a binary relation on A which is reflexive, symmetric, and antisymmetric (so R satisfies all 3 properties). Prove that - {(a,a) € A x A |a € A}. In other words, R is the diagonal in A x A. Note: The hypotheses here are a little different from what was stated in class. The containment {(a, a) € A × A|a e A} C R follows from the fact that R is reflexive. The containment R C {(a, a) E A × A|a E A} will take more work and uses the other 2 properties.

1. Let A be a set and suppose R is a binary relation on A which is reflexive, symmetric, and antisymmetric (so R satisfies all 3 properties). Prove that - {(a,a) € A x A |a € A}. In other words, R is the diagonal in A x A. Note: The hypotheses here are a little different from what was stated in class. The containment {(a, a) € A × A|a e A} C R follows from the fact that R is reflexive. The containment R C {(a, a) E A × A|a E A} will take more work and uses the other 2 properties.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.7: Relations
Problem 29E: 29. Suppose , , represents a partition of the nonempty set A. Define R on A by if and only if there...
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1. Let A be a set and suppose R is a binary relation on A which is reflexive, symmetric, and antisymmetric (so R satisfies all 3 properties). Prove that R = {(a,a) € A × A |a € A}. In other words, R is the diagonal in A x A. Note: The hypotheses here are a little different from what was stated in class. The containment {(a, a) E A × A |a € A} C R follows from the fact that R is reflexive. The containment R C {(a, a) E A × A |a € A} will take more work and uses the other 2 properties. 2. Define - on R² by (x, y) ~ (u, v) if and only if æ² + y² = u² + v². In class, we saw that this defines a binary relation on R? which is reflexive, symmetric, and transitive, hence it is an equivalence relation. Describe all of its equivalence classes.
Transcribed Image Text:1. Let A be a set and suppose R is a binary relation on A which is reflexive, symmetric, and antisymmetric (so R satisfies all 3 properties). Prove that R = {(a,a) € A × A |a € A}. In other words, R is the diagonal in A x A. Note: The hypotheses here are a little different from what was stated in class. The containment {(a, a) E A × A |a € A} C R follows from the fact that R is reflexive. The containment R C {(a, a) E A × A |a € A} will take more work and uses the other 2 properties. 2. Define - on R² by (x, y) ~ (u, v) if and only if æ² + y² = u² + v². In class, we saw that this defines a binary relation on R? which is reflexive, symmetric, and transitive, hence it is an equivalence relation. Describe all of its equivalence classes.
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