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
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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![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7a6f4d1-5574-4afc-bfb9-6d15c839bf1a%2Ff0105acc-c12f-4afc-a94b-717a159913a8%2Frw5tmc8_processed.png&w=3840&q=75)
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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