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/sci/ - Science & Math

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9242062 No.9242062 [Reply] [Original]

1. If a conjecture is strong enough does proving it provide any practical benefits?

2. Have any conjectures that we thought was true turn out to be false?

3. How valid are proof by contradictions? How does starting with a premise that X is not Y and then ending up with sheningans a valid proof that X is Y? Why is it ok to do that you can't start your proof of with X = Y when trying to prove X = Y and then using algebra and other math to work X or Y into Y or X.

>> No.9242071

I conjecture that the Riemann hypothesis is undecidable. The practical benefit is that this upsets people.

>> No.9242074

>>9242062
1. Proofs often reveal corollaries or techniques with practical value.

2. Yes.

3. Very valid.

>How does starting with a premise that X is not Y and then ending up with sheningans a valid proof that X is Y?
Because if something implies a falsehood or its own negation, it cannot be true. If it's not true, then it's negation is true.

>Why is it ok to do that you can't start your proof of with X = Y when trying to prove X = Y and then using algebra and other math to work X or Y into Y or X.
What?

>> No.9242082

>>9242074
Like I'm trying to show that 5*4=40/2 for example (juvenile obviously, but I don't want to learn math code tonight and use the log example I had in class today) my teacher said I can't use algebra and work my way down to 20=20 thus 5*4=40/2 since I'm not allowed to start with the premise that the equation is true/

>> No.9242179

bump

>> No.9242297

>>9242062
>How does starting with a premise that X is not Y and then ending up with sheningans a valid proof that X is Y?
This is called proof by contradiction. It relies on that fact that "If X then Y" necessarily implies "If not Y then not X."

>Why is it ok to do that you can't start your proof of with X = Y when trying to prove X = Y and then using algebra and other math to work X or Y into Y or X.
Because "If X then Y" does not imply "If Y then X." Also because the algebra working out could very well be a coincidence.

>> No.9242308

>>9242297
What if you start with X = Y Use algebra to turn X into C and use algebra to turn Y into C. (Either by manipulating the equation algebraically or breaking it apart and independently turning X into C and Y into C

>> No.9242329

>>9242308
If you independently show X=C and Y=C, you have proved X=Y. You can't start with the premise being what you're trying to prove, though.

>> No.9242339
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9242339

>>9242329

So if I am asked to show X = Y I can't use any algebraic manipulation to get into a position where C = C?

If I'm asked to show X = Y can I be like. Suppose X =/= Y and then use algebraic manipulation and get to a point where C=/=C, thus proving X = Y

If so, where is the first one not allowed but the second one is?

>> No.9242387

>>9242339
Using algebraic manipulation tot prove x=y seems valid though. Eg to prove that in a group if an element a has 2 inverses, say b and c then b=c. Like this b=b+0=b+(a+c)=(b+a)+c=0+c=c but you dont start of saying that b=c, because thats what you want to prove. How can you prove something if you take it as a given?

>> No.9242597

You can start by assuming X=Y and arrive to something like X=X or Y=Y and use that as scratch work to then write the same algebra backwards, starting from X=X and work your way to X=Y, as long as you didn't use any nonreversible functions, it should be exactly the same algebra both ways.
Examples:
Show that if x=y, then x+7=y+7
Suppose x+7=y+7. Subtract 7 from both sides and you get x=y. Since addition is reversible, you can use the exact same algebra backwards.

Show that if |x|=|y| then x=y.
Suppose x=y, then x2=y2. Take the square root from both sides and |x|=|y|.
However, in this case, you can't just write the same algebra backwards because the square root isn't reversible and if you apply it to x2, you would get x=y or x=–y, which doesn't prove anything useful.


>>9242297
>>How does starting with a premise that X is not Y and then ending up with sheningans a valid proof that X is Y?
>This is called proof by contradiction. It relies on that fact that "If X then Y" necessarily implies "If not Y then not X."
You're mixing up contrapositive and contradiction.

>> No.9242599

>>9242597
God damn it, I knew I should have used latex. Anyway [math] x2 \equiv x^{2} [/math]