Equivalence Relations in Set Theory: Homework Statement and Solutions

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Homework Help Overview

The discussion revolves around the concept of equivalence relations in set theory, specifically focusing on the properties that define such relations and the implications of these properties on a set A.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand how to construct a proof for the properties of equivalence relations. Some participants question the implications of having an element in A that is not related to any other member, while others explore specific examples of relations and their interpretations.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of equivalence relations and questioning the nature of equivalence classes. There is no explicit consensus, but various lines of reasoning are being examined.

Contextual Notes

Participants are considering specific examples and definitions from a textbook, which may influence their understanding and assumptions about equivalence relations.

imranq
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Homework Statement


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Homework Equations


An equivalence relation on a set A, is for a,b,c in A if:
a~a
a~b => b~a
a~b and b~c => a~c

The Attempt at a Solution


It seems uncomplicated, but I don't know how I would write down a proof. The book I'm using is Topics in Algebra, 1st Edition Herstein
 
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What if there were some element a \in A which wasn't related to any other member of A?
 
In other words, consider A= {1, 2, 3} and the relation is {(1, 1), (1,2), (2,1), (2,2)},
 
HallsofIvy said:
In other words, consider A= {1, 2, 3} and the relation is {(1, 1), (1,2), (2,1), (2,2)},

I don't understand this. I think equivalence classes are generalized equal signs for some property. So (1,1) I understand, but how so for (1,2)?
 

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