give a rigorous proof

[triclino]

given a set with the symbols :

 

" + " for addition

 

" - " for inverse

 

the constants :

 

1

 

0

 

AND the axioms:

 

for all a,b,c : a+(b+c) = (a+b) + c

 

for all a : a+0 = a

 

for all a : a +(-a) =0

 

for all a,b : a+b = b+a

 

Give a rigorous proof of the following:

 

1) 0 is unique

 

2) the inverse of a ,-a is unique

 

3) for all x,y : -( x+y) = -x +(-y)

[timo]

There's a (written or unwritten, I am not sure) policy on sfn that others should not do your homework for you. Try it yourself. If you get stuck or your attempts fail, then ask for help and present what you did/tried so far.

[triclino]

There's a (written or unwritten, I am not sure) policy on sfn that others should not do your homework for you. Try it yourself. If you get stuck or your attempts fail, then ask for help and present what you did/tried so far.

 

I am not a student and this is not homework .

 

if you noticed this a rigorous proof and not simply a proof .next step to rigorous proof is a formal proof.

 

I am simply interested to see how other people apart from my self are approaching the problem

[Bignose]

Proofs by contradiction usually work well for the first two -- that is, assume that there isn't a unique zero and show that assuming that that leads to conclusions that break the axioms, and therefore there is a unique zero. Same with stating with a non-unique inverse.

 

Whether it is homework for a class or not, the rule remains the same. Members can give clue or hints (like the above), but we still don't just do all the work for someone else. If you want to post your proof or a few different ones in the interest of discussing them, that's fine. But, the forum justifiably so will always err on the side of caution. And the cautious approach here is that this looks like homework, so I really hope that no one just posts answers.

[triclino]

Proofs by contradiction usually work well for the first two -- that is, assume that there isn't a unique zero and show that assuming that that leads to conclusions that break the axioms, and therefore there is a unique zero. Same with stating with a non-unique inverse.

 

Whether it is homework for a class or not, the rule remains the same. Members can give clue or hints (like the above), but we still don't just do all the work for someone else. If you want to post your proof or a few different ones in the interest of discussing them, that's fine. But, the forum justifiably so will always err on the side of caution. And the cautious approach here is that this looks like homework, so I really hope that no one just posts answers.

 

 

I am sorry, i should have pointed out that rigorous proof means ,a proof in which every line is justified by the appropriate axiom.

 

The proof by contradiction that you suggest ,i do not know how to do , so please if you wish show more details

[the tree]

Proofs by contradiction are fairly simple, first presume the negate of what you're trying to prove, such as the existence of two distinct zeros. Then work that into a statement (in the same fashion as a direct proof) that contradicts one of the axioms that you've given.

[triclino]

So we want to prove that:

 

There exists a unique y for all x such that: x+y =x

 

And now using contradiction we must negate the above statement

 

AND the question now is what is the negation of the above statement??

 

 

In quantifier form the above statement is:

 

[math]\exists y!\forall x[ x+y=x][/math]...... ([math]\exists !y[/math] means there exists a unique y) , which by definition is equivalent to:

 

 

[math]\exists y\forall x[x+y = x][/math] AND [math]\forall y\forall z[\forall x( x+y=x)\wedge\forall x(x+z=x)\Longrightarrow y=z][/math],

 

which in words is:

 

there exists a, y for all x such that :x+y =x

 

AND

 

FOR all y,z :if for all x ( x+y=x) and for all x(x+z=x) ,then y=z

 

Now to negate the above is not an easy task.

 

That is why i asked Bignose to give more details when he suggested the proof by contradiction.

 

 

However if anybody insists in a proof by contradiction ,i think, she/he must show more details

[timo]

I'd spontaneously not try to approach it by using contradiction. I'd show that if there are two inverses (or neutral elements) a and b, then a=b. But thinking about it that might be what Bignose meant with a proof by contradiction (the result it is kind of a contradiction to the assumption but I do not get where a violation of the axioms would come in).

[D H]

So we want to prove that:

 

There exists a unique y for all x such that: x+y =x

 

And now using contradiction we must negate the above statement

 

AND the question now is what is the negation of the above statement??

Per your axioms, there already exists one number, zero, that satisfies the above. Therefore the negation of the uniqueness is that there exists some x such that there exists some y that is not equal to zero such that x+y=x.

 

And then we are stuck because your axioms do not define the concept of inequality.

[Bignose]

I didn't insist on anything, I was trying to give a hint, and I still am not going to provide a detailed proof because I don't like spoon feeding anybody. If you really are driven to find someone else to do the work for you, these proofs can be found in the literature (and no, I'm not going to tell you which book).

[triclino]

I didn't insist on anything, I was trying to give a hint, and I still am not going to provide a detailed proof because I don't like spoon feeding anybody. If you really are driven to find someone else to do the work for you, these proofs can be found in the literature (and no, I'm not going to tell you which book).

 

 

 

How can you spoon feed me ,if the rigorous proofs of the above exist in books as you claim.

 

All you had to do from the very beginning was to refer me to the particular book(s) ,instead of suggesting a method which you cannot follow to the very end

[the tree]

He could, obviously he could, he's a perfectly competent mathematician - but what good would it do you? The rules are not there just for the lulz, they are there for a reason.

 

The fact is that BN is trying to act in your interest and you're throwing it in his face - not cool.

[triclino]

Please!!! my intentions are not to throw anything at anybody's face.

 

If you read my posts again in one of them i said i could not do the proof by using contradiction and i asked for more details.

 

Yet i cannot do the proof by contradiction,and if anybody else can,please do so

[triclino]

Bignose ,i looked uppon many books of linear and abstract algebra and none of them gives a proof by contradiction.

 

So please be so kind ,since you suggested and i pressume you know,as to produce,if you wish, the proof by contradiction

[Bignose]

I am not going to give you the proof, one more time I am not going to do your work for you. However, I know for a fact that a proof by contradiction is given in Shilov's Elementary Real and Complex Analysis.

[triclino]

I am not going to give you the proof, one more time I am not going to do your work for you. However, I know for a fact that a proof by contradiction is given in Shilov's Elementary Real and Complex Analysis.

 

 

Producing a particular proof YOU DO NOT HELP ME with my work because the contradictory proof it never crossed my mind,untill you mentioned it,

 

YOU SIMPLY add valuable knowledge to the forum

[iNow]

YOU SIMPLY add valuable knowledge to the forum

 

Which Bignose does with practically every single post he makes here. I strongly suggest that you are likely to find yourself clinging to the minority opinion if you dislike his methods.

[Bignose]

Producing a particular proof YOU DO NOT HELP ME with my work because the contradictory proof it never crossed my mind,untill you mentioned it,

 

YOU SIMPLY add valuable knowledge to the forum

 

Nice.

 

The rule on the forum about not spoon feeding work to other people has been mentioned several times, and yet you insist on demanding it. I even found the book that I know a proof exists in, have you even looked in the book I cited? I mean, for goodness sakes, the proof is even on the first 10 or 15 pages or so, you don't even have to read the whole thing! Learning how to look things up for yourself is a valuable skill, too. Not everything is going to be able to be gotten from members on a forum.

 

Edited:

 

I guess you did find someone to spoon-feed you your answer, congratulations:

 

http://www.mathhelpforum.com/math-help/discrete-mathematics-set-theory-logic/94153-proof-contradiction.html

 

So, since you already had that in hand, why the need to come back and complain?!? Funny, funny, stuff.

Edited July 3, 2009 by Bignose

[SH3RL0CK]

Nice.

 

The rule on the forum about not spoon feeding work to other people has been mentioned several times, and yet you insist on demanding it. I even found the book that I know a proof exists in, have you even looked in the book I cited? I mean, for goodness sakes, the proof is even on the first 10 or 15 pages or so, you don't even have to read the whole thing! Learning how to look things up for yourself is a valuable skill, too. Not everything is going to be able to be gotten from members on a forum.

 

Edited:

 

I guess you did find someone to spoon-feed you your answer, congratulations:

 

http://www.mathhelpforum.com/math-help/discrete-mathematics-set-theory-logic/94153-proof-contradiction.html

 

So, since you already had that in hand, why the need to come back and complain?!? Funny, funny, stuff.

 

 

Strange. Very strange. He seems to have spent more time demanding to be spoon-fed the answer than it would have taken to simply figure out the solution for himself. And for all that work, he now has the answer (which I doubt he really understands) to one problem, rather than the ability to solve many problems.

[mooeypoo]

triclino, welcome to ScienceForums.

 

Your question sounds like it's for homework, but regardless of whether or not it is, we're not usually doing "rigorous proof" for other people. If you want others to show you how to approach it, then just read the posts above (because they did tell you) and rephrase the question. No one here will just sit down and solve the problem (that is already solved) for you.

 

If your intension is to verify that what you've already done is correct, then the least you can do is post some of it here for others to review.

 

 

In short, if you want assistance, you will need to cooperate.

Other than that, I urge everyone on the thread to remain patient and curteous and remember that we are here for a scientific discussion, and not some geusswork practice of the personal reasons someone asked a question.

Everyone should stick to the topic at hand, and the OP should start cooperating.

 

~moo

[triclino]

Nice.

 

The rule on the forum about not spoon feeding work to other people has been mentioned several times, and yet you insist on demanding it. I even found the book that I know a proof exists in, have you even looked in the book I cited? I mean, for goodness sakes, the proof is even on the first 10 or 15 pages or so, you don't even have to read the whole thing! Learning how to look things up for yourself is a valuable skill, too. Not everything is going to be able to be gotten from members on a forum.

 

Edited:

 

I guess you did find someone to spoon-feed you your answer, congratulations:

 

http://www.mathhelpforum.com/math-help/discrete-mathematics-set-theory-logic/94153-proof-contradiction.html

 

So, since you already had that in hand, why the need to come back and complain?!? Funny, funny, stuff.

 

The proof written in the MHF that you refered me to,

 

 

IS NOT A PROOF BY CONTRADICTION

[D H]

The proof written in the MHF that you refered me to,

 

 

IS NOT A PROOF BY CONTRADICTION

Of course they are. Both of those proofs of start with assumptions to the contrary, that zero and the additive inverse are not unique. Both proofs proceed to show that these assumptions lead to a contradiction.

[triclino]

triclino, welcome to ScienceForums.

 

Your question sounds like it's for homework, but regardless of whether or not it is, we're not usually doing "rigorous proof" for other people. If you want others to show you how to approach it, then just read the posts above (because they did tell you) and rephrase the question. No one here will just sit down and solve the problem (that is already solved) for you.

 

If your intension is to verify that what you've already done is correct, then the least you can do is post some of it here for others to review.

 

 

In short, if you want assistance, you will need to cooperate.

Other than that, I urge everyone on the thread to remain patient and curteous and remember that we are here for a scientific discussion, and not some geusswork practice of the personal reasons someone asked a question.

Everyone should stick to the topic at hand, and the OP should start cooperating.

 

~moo

 

Mr Moderator ,thank you for your wellcoming to the forum and your valuable advice but,

 

a proof that cannot be found in books we cannot call homework.

 

The forum that Bignose suggested ( MHF) ,have two proofs ,one for the uniqness of zero and one for the uniqness of the inverse ,which are not proofs BY CONTRADICTION.

 

The book that Bignose suggested it is difficult to get.

 

 

SO FOR ONE MORE TIME I BEG THE WHOLE FORUM IF ANYONE KNOWS THE PROOF BY CONTRADICTION,PLEASE WRITE IT HERE.

 

Note: in my post #7 i pointed out the difference between a direct and indirect proof ( = contradiction),which i suspect many people mix up,for that particular proof ,that is

[D H]

The forum that Bignose suggested ( MHF) ,have two proofs ,one for the uniqness of zero and one for the uniqness of the inverse ,which are not proofs BY CONTRADICTION.

What in the world makes you think those are anything but proof by contradiction?

[mooeypoo]

Mathematical proof requires some analytical skills, it does not involve magic or conjuring, so it is possible to reach at it without it previously being in a book. In fact, many *GOOD* math professors will ask you to come up with such proof to train your analytical mind, to see if you know how to approach it, to at least try.

 

Of course it can be homework. It can be the best KIND of homework, the kind that actually teaches you something. If it was already in a book, it wouldn't be good homework, it would be good copying.

 

That said, in order for you to even BEGIN solving such a question - homework or not - you have to at least start somewhere. Show us your style, show us what you're thinking, anything. This will help you, because you will get to start it out. And this will help us because we will see your style and have a better grasp on how to help you.

 

Either way, you have to start cooperating here, triclino. Start listening to the replies people put for you and stop being so insistant that they spoon-feed you. I'm sure you're much more intelligent than that, and can do more than copy off the screen (or off some book).

 

If you think we misunderstood your question, then please rephrase it. Otherwise, I recommend you just start reading people's answers and cooperating back.

 

~moo