My Math Forum Abstract Algebra: Groups and Subgroups Problem

 Abstract Algebra Abstract Algebra Math Forum

 October 19th, 2012, 06:22 AM #1 Newbie   Joined: Oct 2012 Posts: 1 Thanks: 0 Abstract Algebra: Groups and Subgroups Problem Let R be the set of real numbers. Determine whether the relation H on R defined as xHy <=> y=kx for an integer k, is (a) reflexive (b) symmetric (c) transitive Is H an equivalence relation?
 October 19th, 2012, 07:44 AM #2 Senior Member   Joined: Nov 2011 Posts: 100 Thanks: 0 Re: Abstract Algebra: Groups and Subgroups Problem What have you tried so far? What can you tell me about the relation between some relation satisfying a,b,&c and being an equivalence relation?
 October 20th, 2012, 01:38 PM #3 Senior Member   Joined: Jul 2011 Posts: 227 Thanks: 0 Re: Abstract Algebra: Groups and Subgroups Problem (a) It's clear that $x= kx \Rightarrow k=1 \in \mathbb{Z}$ therefore the relation is reflexive. (b) We have $xHy \Leftrightarrow \exists k \in \mathbb{Z}: y=kx \Leftrightarrow x = k^{-1}y$ but $k^{-1} \notin \mathbb{Z}$ (only if $k=1$ or $k=-1$ but the statement has to be true for every $x,y \in \mathbb{R}$) thus not symmetric. (c) We have $xHy \Leftrightarrow \exists k_1 \in\mathbb{Z}: y= k_1x$ and also $yHz \Leftrightarrow \exists k_2 \in \mathbb{Z}: z=k_2y$ thus $z= k_2y = k_2(k_1x) = (k_2k_1)x$ and because $k_2k_1 \in\mathbb{Z}$ we have $xHz$ so the relation is transitive. It's not an equivalence relation because only (a) and (c) are satisfied.

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