Quest for equality
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From: Hero on 17 Jun 2010 11:24 Zdislav wrote: > Hero wrote: > > Is there a collection, assemblation, congregation or whatsoever > > of different objekts, which have equal value? > > > { 12 / 6, 5 - 3 , 6 / 3, 1 + 1 } has > > - per definition of a mathematical set - only one element. > > So how can we express this bunch of terms? > I see there a set of strings of symbols. > Tacitly, it is accompanied by a function called "evaluation". > > (Technicalities need to be provided, such as the domain and > the codomain of that function.) > > Then indeed, the image of that set of strings under the > evaluation function has one element. > > Good enough? Thanks for Your words. Actually, what I was looking for, the multiset, was given by Gerry. With friendly greetings Hero
From: Hero on 17 Jun 2010 11:26 Gerry wrote: > Hero wrote: > > Is there a collection, assemblation, congregation or whatsoever > > of different objekts, which have equal value? > > > { 12 / 6, 5 - 3 , 6 / 3, 1 + 1 } has > > - per definition of a mathematical set - only one element. > > So how can we express this bunch of terms? > > There is something called a multi-set, which is like a set except it's > allowed to have repeated elements. Informally, such a thing may be > referred to as a bag. That is exactly I was looking for. At first I thought about a tuple or a sequence ( 12 / 6, 5 - 3 , 6 / 3, 1 + 1 ) but here the position of the elements is an unnecessary extra. So { 1; 2 ; 1 + 1 } = { 2; 1 + 1; 1 } answers my question. (I write a multiset with semicolon, so it has a clear distinction with a ordinary set.) With this preliminary let's procced in the quest for equality: Does the field of rationals with its addition and multiplication ( Q|, +, * ) has exactly one neutral element for multiplication, the 1 ? The 'problem' is: if there exists only one single neutral element ( the 1 ), how can one add: 1 + 1? With friendly greetings Hero
From: Gerry Myerson on 17 Jun 2010 18:35 In article <1046951a-c295-4931-b060-8c831084b107(a)g19g2000yqc.googlegroups.com>, Hero <Hero.van.Jindelt(a)gmx.de> wrote: > Gerry wrote: > > �Hero wrote: > > > Is there a collection, assemblation, congregation or whatsoever > > > of different objekts, which have equal value? > > > > > { 12 / 6, 5 - 3 , 6 / 3, 1 + 1 } has > > > �- per definition of a mathematical set - only one element. > > > So how can we express this bunch of terms? > > > > There is something called a multi-set, which is like a set except it's > > allowed to have repeated elements. Informally, such a thing may be > > referred to as a bag. > > That is exactly I was looking for. > At first I thought about a tuple or a sequence > ( 12 / 6, 5 - 3 , 6 / 3, 1 + 1 ) > but here the position of the elements is an > unnecessary extra. > So > { 1; 2 ; 1 + 1 } = { 2; 1 + 1; 1 } > answers my question. > (I write a multiset with semicolon, so it has a > clear distinction with a ordinary set.) > > With this preliminary let's procced > in the quest for equality: > > Does the field of rationals with its addition and multiplication > ( Q|, +, * ) has exactly one neutral element for multiplication, > the 1 ? > > The 'problem' is: > if there exists only one single neutral > element ( the 1 ), how can one add: > 1 + 1? Ha. Reminds me of the theory that there's only one electron in the universe, http://en.wikipedia.org/wiki/One-electron_universe -- Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Hero on 18 Jun 2010 11:03 Gerry wrote: > Hero wrote: > > > Does the field of rationals with its addition and multiplication > > ( Q|, +, * ) has exactly one neutral element for multiplication, > > the 1 ? > > > The 'problem' is: > > if there exists only one single neutral > > element ( the 1 ), how can one add: > > 1 + 1? > > Ha. Reminds me of the theory that there's only one electron > in the universe,http://en.wikipedia.org/wiki/One-electron_universe > That is a strange theory. Now, I often heard about the uniqueness of the neutral element. "There is exactly one neutral element." http://iamwww.unibe.ch/~halbeis/4students/gtln/sec1.pdf "As a result, we can speak of the identity element of { G , # } rather than an identity element." http://en.wikipedia.org/wiki/Elementary_group_theory "In particular, there can never be more than one two-sided identity. If there were two, e and f, then e * f would have to be equal to both e and f." http://en.wikipedia.org/wiki/Identity_element So, in 1 + 1, is 1 here added "to itself"? With friendly greetings Hero
From: Hero on 19 Jun 2010 11:02
Hero wrote: > So, in 1 + 1, is 1 here added "to itself"? Most of you know this proof and as it is so convincing, the accompanying commentary is digested without chewing: Proposition 1.1. Let (G; o ) be a group, then there is exactly one neutral element and each element of G has exactly one inverse. Proof. Let e; ~e element of G be neutral elements of (G; o ). Thus, for every x element of G we have x o ~e = e o x = x, and therefore, e = e o ~e = ~e, " " ~e neutral e neutral and hence, there is exactly one neutral element. http://iamwww.unibe.ch/~halbeis/4students/gtln/sec1.pdf Give it a thought. It states, if there are two neutral elements, then they are equal. Nowhere is forbidden, that there are more equal elements. The 'conclusion' : "there is exactly one neutral element" is a confusion of identy with equality. And with rational numbers there have to be several ones, several twos, several twelves, ..., otherwise terms like 1 + 1 can not be constructed. With friendly greetings Hero |