
Abstract Algebra Abstract Algebra Math Forum 
 LinkBack  Thread Tools  Display Modes 
October 10th, 2014, 10:53 AM  #1 
Member Joined: Jun 2011 From: California Posts: 82 Thanks: 3 Math Focus: Topology  Equivalence Relation Question
Hi all! I'm reading right now about the quotient topology and quotient spaces, and I've come across an example that uses this statement: "On $\mathbb{R}^2$, let $\equiv$ be the equivalence relation generated by $(x, y) \equiv (x, y)$." I'm not familiar with this treatment of equivalence relations; I'm used to a book explicitly defining an equivalence relation in the way of "$(x, y) \equiv (a, b)$ if and only if . . ." . Do I just treat this as "The equivalence relation on $\mathbb{R}^2$ such that every point $(x, y)$ is related to the point $(x, y)$"? If not, how do treat this? Having a Master's, I'm well acquainted with equivalence relations, but I'm hoping I wasn't just spacing out as an undergrad on the day equivalence relations were taught in my introductory proofs class. Last edited by matthematical; October 10th, 2014 at 10:57 AM. 
October 10th, 2014, 03:46 PM  #2 
Senior Member Joined: Dec 2013 From: Russia Posts: 327 Thanks: 108  This means that $\equiv$ is the reflexive, symmetric and transitive closure of the given relation, i.e., the smallest equivalence relation containing the given one. In this case, one only has to take the reflexive closure, I believe.


Tags 
equivalence, question, relation 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Equivalence relation  tom33  Algebra  3  January 17th, 2014 05:30 PM 
Equivalence Relation  Taladhis  Abstract Algebra  2  February 11th, 2013 09:20 AM 
Equivalence Relation  jrklx250s  Real Analysis  3  December 7th, 2011 11:42 AM 
Equivalence Relation question  jstarks4444  Number Theory  1  October 30th, 2011 07:39 PM 
equivalence relation  tinynerdi  Abstract Algebra  1  January 11th, 2010 10:24 AM 