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proof!


sysD
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incorrect. any takers?

Incorrect. Dr Rocket is correct. The identity relationship is reflexive by definition.

 

EDIT: its not what you think

Oh really. Perhaps you should tell us what you think then. What set, what axioms, and what definition of equality are you using that makes you think that "it's not what you think"?

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1=1 and not 1!=1 then x!=x is also wrong.

 

 

We have not been informed what kind of objects x are. They may not be numbers. ;)

 

DrRocket is right there is nothing to prove. The statement x=x is the reflexive relation of "equals". We understand this as being an intrinsic property of "equals".

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I'm serious guys. Its very counter-intuitive.

 

I can't give out any hints though; I know someone here is smart enough to prove it.

 

OH Crap. Sorry, I posted the wrong equation.

I can't edit the OP so here it is:

________________________________________________________________________

 

 

Prove that:

 

x=/=x

 

x is in the set of Real Numbers

 

 

 

 

[couldnt figure out the math script notation for "does not equal..." if a mod would like to help me out i'd appreciate it.]

Edited by sysD
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[math]x\neq x[/math]?

 

 

A valid proof of a patently false assertion can only be made using an inconsistent set of axioms, in which case any assertion is both true and false.

 

Inconsistent axiom systems are not very interesting.

 

The only alternative was alluded to by DH, namely do something stupid and call it clever.

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Oh please. We don't need another stupid division by zero "proof" to show that 1=2 or x≠x.

 

You're close, but that's not it. I promise you its something very simple and every day - yet I don't think many would think to apply it here.

 

 

 

 

A valid proof of a patently false assertion can only be made using an inconsistent set of axioms, in which case any assertion is both true and false.

 

Inconsistent axiom systems are not very interesting.

 

The only alternative was alluded to by DH, namely do something stupid and call it clever.

 

Try it out, as a thought experiment. You may be surprised.

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You're close, but that's not it. I promise you its something very simple and every day - yet I don't think many would think to apply it here.

You cannot prove x≠x. It is nonsense. Mathematics is extremely intolerant of nonsense.

 

It is of course possible to arrive at a result by means of some calculation. However, this does not mean you have proven that x≠x. It instead means that

- You made a mistake somewhere along the way in getting to that result, or

- One or more of your assumptions is invalid.

 

This latter possibility leads to a very powerful mathematical tool for proving or disproving hypotheses, powerful precisely because mathematics is extremely intolerant of nonsense. It's called proof by contradiction.

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I reckon if you really trawled through the forums then you'd find a handful.

 

Agreed. I was referring to this particular thread. (Added in edit: And at the time I wrote it. I am not responsible for the effect on the truth of my assertion that may be caused by any latecomers.)

 

If you were to trawl Philosophy your net might not take the load.

Edited by DrRocket
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When you end an actual mathematical proof, don't you end it with "thus" followed by an equation such as "x=x"? I don't think you can go any further. I think any other proof would be needlessly adding things.

x=x because there isn't anything saying it isn't, x is by the identity of x equal to x.

Edited by questionposter
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x=x because there isn't anything saying it isn't, x is by the identity of x equal to x.

 

Really it comes down to the basic fundamental and natural properties of equivalence relations.

 

Very loosely, an equivalence relation, lets say on elements of some set, gives us a way of saying if the elements are "alike" or not. We can then break-up the set into smaller collections of equivalent elements. These are the equivalence classes. Don't panic, all I am really saying is we can define in many many ways how two elements can in essence be the "same" and this will of course depend on the context.

 

Anyway, I think we would end up being very confused if we said that "x is not like x with respect to some property that x has". :huh:

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