Path: dp-news.maxwell.syr.edu!spool.maxwell.syr.edu!drn.maxwell.syr.edu!news.maxwell.syr.edu!newsfeed.icl.net!newsfeed.fjserv.net!nx01.iad01.newshosting.com!newshosting.com!207.115.63.142.MISMATCH!newscon02.news.prodigy.com!newscon06.news.prodigy.com!prodigy.net!border1.nntp.dca.giganews.com!local01.nntp.dca.giganews.com!nntp.comcast.com!news.comcast.com.POSTED!not-for-mail NNTP-Posting-Date: Sun, 07 Aug 2005 22:10:46 -0500 From: "VC" Newsgroups: comp.databases.theory References: <1122847137.142084.67550@g44g2000cwa.googlegroups.com> <%WdHe.76187$s54.28208@pd7tw2no> <1122960074.359597.49310@g14g2000cwa.googlegroups.com> <3OJHe.9182$6f.1603@newsread3.news.atl.earthlink.net> <1122991673.984098.255740@z14g2000cwz.googlegroups.com> <42ef9948$0$24030$ed2619ec@ptn-nntp-reader01.plus.net> <_tGdnYL8uLqmb3LfRVn-uQ@comcast.com> <42f12830$0$24039$ed2619ec@ptn-nntp-reader01.plus.net> <7sOdnZvcEttiHGzfRVn-1w@comcast.com> <1123167475.395012.72690@o13g2000cwo.googlegroups.com> <42f4cf30$0$91514$ed2e19e4@ptn-nntp-reader04.plus.net> <1123346399.004151.101270@g47g2000cwa.googlegroups.com> <42f50fc1$0$24006$ed2619ec@ptn-nntp-reader01.plus.net> Subject: Re: The naive test for equality Date: Sun, 7 Aug 2005 23:10:46 -0400 X-Priority: 3 X-MSMail-Priority: Normal X-Newsreader: Microsoft Outlook Express 6.00.2900.2180 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2900.2180 X-RFC2646: Format=Flowed; Original Message-ID: Lines: 36 NNTP-Posting-Host: 24.91.127.140 X-Trace: sv3-LwICQYhKwJKdQYpNMZBGHzf9THPIPvgbJpLdTeDOitu0S3C4F2DXF6rY/qrR5j6ndCgPZhjsw7UTQa5!q3JW0GPp4OqjeTnF1MfNYecEJN6JKg0XDApOcinFsPeKRYGYH6X0Js4XP4YQz4Pqyp5N31+DWQ== X-Complaints-To: abuse@comcast.net X-DMCA-Complaints-To: dmca@comcast.net X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.32 Xref: dp-news.maxwell.syr.edu comp.databases.theory:32742 "Paul" wrote in message news:42f50fc1$0$24006$ed2619ec@ptn-nntp-reader01.plus.net... >... The way I see it, an equivalence relation > *defines* what we mean by equality with respect to a given structure. > > So for example you start with expressions of the form "x/y", with x and > y integers (y!=0) > > Now to begin with, "1/2" != "2/4" > > But you create an equivalence relation as VC described, which is > basically grouping certain integer pairs together to create a different > structure. And you use this equivalence relation to *define* what you > mean by "equality" on your new structure. So [1/2] = [2/4]. But > conventionally you drop the square brackets indicating the equivalence > class and write 1/2 = 2/4, which maybe confuses things though. Right... > > So for the rational numbers, you have equality but the corresponding > equivalence classes on ZxZ *don't* have cardinality 1 Not quite right. In the case of rationals equality, you treat the equivalence class, as a whole, as a single element. E.g, for integers you'd say 2=2; for rationals you'd say [5/10] = [1/2], no difference really since both [5/10] and [1/2] is the *same* element. In other words, your *equality* relation pair would be, say, for integers (1,1) and for rationals (E_half, E_half), where E_half = {1/2, 2/4,, 5/10, ..} etc. > > Paul.