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January 8

Group Axioms

My lecturer stated (without proof, naturally) that if we take the following defintion of a group:

A set S with a binary operation " $\cdot$ " which maps $S\times S\rightarrow S$ and obeying the following,

- Associativity: $a\cdot (b\cdot c)=(a\cdot b)\cdot c$

- Identity: $a\cdot e=a$

- Inverse: $a\cdot a^{-1}=e$

(where all the elements exist and belong to S etc.)

Then $e\cdot a=a$ follows as a theorem.

Now, in the books that I've seen and the wikipedia article this property is included in the definition of the identity. So, the question for any helpful RefDeskers is: can this be shown from the above or must it be assumed? 163.1.176.253 (talk) 00:29, 8 January 2009 (UTC)[reply]

I guess they mean $e\cdot a=(a\cdot a^{-1})\cdot a=a\cdot (a^{-1}\cdot a)=a\cdot e=a$ . First one has to prove $a^{-1}\cdot a=e$ however, which is another little trick. I have to confess that I do not see the point of this making the axioms as ecomonical as possible (in this case). C'm on, we do not have to pay for them. I prefer the more simmetrical ones :) --PMajer (talk) 00:55, 8 January 2009 (UTC)[reply]

The article, elementary group theory, might be helpful. --Point-set topologist (talk) 09:59, 8 January 2009 (UTC)[reply]

Yes, thank you. There's a very relevant section on alternative axioms providing the proof. 163.1.176.253 (talk) 10:37, 8 January 2009 (UTC)[reply]

(ec) Let's start from

a\cdot e=a

. For

a=e\,\!

it gives us

e\cdot e=e

. We replace the first

e\,\!

with

a\cdot a^{-1}

and get

(a\cdot a^{-1})\cdot e=e

and by associativity

a\cdot (a^{-1}\cdot e)=e

. I'm not sure, however, if this is enough to conclude that

(a^{-1}\cdot e)=e

.... --CiaPan (talk) 10:46, 8 January 2009 (UTC)[reply]

It certainly isn't - if that were true then you would have a.e=e, which only holds in the trivial group. --Tango (talk) 15:52, 8 January 2009 (UTC)[reply]

I think that he was being sarcastic. --Point-set topologist (talk) 17:02, 8 January 2009 (UTC)[reply]

That would seem odd. I think a simple mistake is more likely. (That last e should probably be an a^-1, in which case it isn't quite enough - it uses the fact that if both left and right inverses exist then they are equal and unique, but that needs proving.) --Tango (talk) 19:35, 8 January 2009 (UTC)[reply]

Original research and WP:OR

This is not really a mathematics question but I thought that I would post this here because it is relevant. Most of the time, the questions here are either trivial or of the same difficulty as a textbook exercise. Sometimes the questions are simply posted out of interest and the answer depends on opinion. But is there a possibility that someone asks a problem here that is publishable and includes some of his/her findings with the question? I remember seeing such problems here before and it is possible that such a thing can be done by an inexperienced mathematician (someone who is learning mathematics and is not really familiar with reading journals; for example, a university student). I think it would be appropriate to include, in the guidelines at the top of the page, that people should consider this when asking a question. Any opinions (and how one can add to the guidelines at the top if people agree)?

Thanks!

--Point-set topologist (talk) 13:37, 8 January 2009 (UTC)[reply]

This should be on the talk page. Algebraist 13:39, 8 January 2009 (UTC)[reply]

Thanks for the quick reply but which talk page? --Point-set topologist (talk) 13:42, 8 January 2009 (UTC)[reply]

WT:Reference desk. Algebraist 13:44, 8 January 2009 (UTC)[reply]

Thanks. I posted the message there. Any opinions would be greatly appreciated. --Point-set topologist (talk) 14:45, 8 January 2009 (UTC)[reply]

Seeking a definition of "Positive Terms"

The linguist Ferdinand de Saussure in his Course in General Linguistics states that "...language is a system of differences..." He goes on to state:

"Even more important:a difference generally implies positive terms between which the difference is set up; but in language there are only differences without positive terms."

Somehow, the clause "a difference generally implies positive terms between which the difference is set up" sounds to me like it refers to some sort mathematical idea.

I would appreciate any comments which might shed light on the meaning or references of this clause. Thwap (talk) 15:18, 8 January 2009 (UTC)[reply]

As far as I can see, it has nothing to do with mathematics. I'd understand "positive" here as "factually existing", in a similar sense as in positive statement, positive science or positivism. Anyway, you may have better luck at the language reference desk when it comes to interpreting de Saussure. — Emil J. 16:09, 8 January 2009 (UTC)[reply]

I've posted this also in the Language Ref Desk but get the same type of responses.

When I consider the original context of the statement and the phrase "a difference generally implies positive terms", the operative word being "difference", there is nothing specific to language being stated here. I get the sense that the author is referring to some kind of mathematical operation used to derive differences whereby is matters whether the terms are positive or negative.68.157.93.254 (talk) 20:40, 8 January 2009 (UTC)[reply]

More context is needed in order to decipher de Saussure. Bo Jacoby (talk) 21:18, 8 January 2009 (UTC).[reply]

The statement occurs in "A Course in General Linguistics" on page 118, in the section entitled "The Sign as a Whole", near the beginning of the first paragraph. The document can be found here:[1]Thwap (talk) 11:44, 9 January 2009 (UTC)[reply]

He is not trying to refer to any sort of higher mathematics. The wikipedia article on positive statement is about the term in economics, but rather one should use 7th definition on wikt:positive. There is something in logic and philosophy (related to positivism) that tries or tried to distinguish between positive and negative statements. "This is a dog." is positive, and "This is not a dog." is negative. The reference to differences may be to the idea that a difference, as a "lack", is a negative statement associated to two positive statements. "I had 20 dollars yesterday. I have 10 dollars today. I lack 10 dollars today." The first might be called positive, the second is positive, and the third might be called negative.

I learned about this stuff in a modal logic course, and one of the tasks was to decide if a given collection of symbols was a positive statement or a negative statement (I think the answer was, there is no such decision procedure since the law of the excluded middle does occasionally hold).

At any rate, the point of p116-118 is that the symbols used in language are arbitrary because their primary purpose is not to carry meaning, but rather to be distinguished from each other. The mathematical version of this is called coding theory. It does not matter what bit-strings are used to encode a message, only that it can be reliably decoded. Some Chinese characters consist of two symbols, one that refers to pronunciation and one to meaning, but in some sense the pronunciation part is to distinguish from other words that relate to the same meaning, and the meaning part is to distinguish it from other words that have the same pronunciation. The fact that both are sometimes related to pictures of real things has by this time become only a mnemonic device.

Some people disagree with this sort of thing, and think that the letters, numbers, sounds, etc. we use to communicate have intrinsic meaning. The only easy examples I know of these are somewhat silly or old-fashioned: divine language as in, before the tower of babel meaning and symbol were the same, only after were they separate and confusing; Pseudoscientific metrology#Charles Piazzi Smyth and the Egyptian inch. Probably the textbook here is trying to clearly state that in its analysis of language, there is no "mystical" meaning in language, and one does not need to examine the symbols themselves, only how they differ from other symbols (or earlier versions of the same symbol, etc.).

At any rate, I think the language ref desk answer is sufficient to understand the passage. When a linguist refers to mathematics by analogy, the linguists should explain, not the mathematicians. To a mathematician, it just seems like an expression of the basic explanation of the applications of coding theory to communication. JackSchmidt (talk) 19:12, 9 January 2009 (UTC)[reply]

Thanks for your help.Thwap (talk) 11:22, 10 January 2009 (UTC)[reply]

Question relating to the discriminant

Im given a quadratic equation $x^{2}+kx+k=0$ and Im asked to write down the discriminant in terms of k. I did this and got $k^{2}-4k$ .

Then Im asked to find the set of values k can take so I did the following:

$k^{2}-4k<0$ $k(k-4)<0$ Im pretty sure this is all right so far but I dont know what to do next. Is it that the 2 possible solution are k<0 and k<4 so overall k<4 or what? --212.120.247.244 (talk) 21:03, 8 January 2009 (UTC) perhaps you are searching for those value of k ,for which the given equation has real roots,for this take k(k-4) > 0 instead of k(k-4) < 0. and check the values of k(k-4) " any value in this range " , in intervals k<0 , 0 < k < 4 and k > 4 then you will get the required values . —Preceding unsigned comment added by Khubab (talk • contribs) 21:28, 8 January 2009 (UTC)[reply]

I'm afraid that this question is really meaningless. For a start, k can be anything; I don't see the problem with complex numbers. If you want real roots, then k is less than or equal to 0 or k is greater than or equal to 4. This is easy to see because k * (k-4) is greater than or equal to 0, when either both factors are greater than 0, both less than 0 (the product of two negative numbers is positive), or one of the factors is 0.

Also note that the discriminant can be precisely 0 for real roots. Hope this helps. --Point-set topologist (talk) 22:00, 8 January 2009 (UTC)[reply]

to find the set of values k can take in order to what? You forgot to turn the page, maybe? ;) --84.220.230.137 (talk) 22:14, 8 January 2009 (UTC)[reply]

I imagine there is an unstated assumption that k is real, so the question is intended to be "what is the range of the function f:R->R, f(k) = k² − 4k". Gandalf61 (talk)