Understanding Proofs in Math: Tips & Book Suggestions

[hellbike]

I enjoy math, but when i have to proof something - i find it confucing.

I don't know how strict proof should be.

If proof ends with 17<18 , it this really end? Why shouldn't i proof that 17<18 is actually true?

I never know what sentences can i treat as obvious, and which one i have to proof.

This is same about proofs i find in books - there are conclusions that are treaten as obvious and not being proved. And i don't know when i can assume something is obvious.

i could use some tips, or book suggestions.Description of my problem may be confusing, but i think is common and i will be understood. if not, i shall provide some examples.

 

[Char. Limit]

Why should you have to prove that seventeen is less than eighteen?

If you must, subtract seventeen from both sides. Then you get 0<1, and you can't have more strict than that.

I think that once either the two sides are shown to be equal, or something arises that's just numbers, you don't need to go further.

 

[hellbike]

this is how proof is limit existence looks in spivak's book -


for every q exists w that meets requirement - f(q) < g(w)

for example, for every q there is w that : 1/q < w

I don't see that being obvious.

I see this it is true and i can provide finite number of examples, but that don't prove anything.

Someone could ask why above sentence(1/q<w) is true and i don't know answer for that.

 

[Char. Limit]

But that's not true. I can name one q that doesn't work: zero.

 

[hellbike]

1/0 is not a number, i think.

I'm not sure how does it works, but i though that if we say:

for every x in R

and then put x in some function, x must be in domain of that function and we don't have to mention that.

But if that's not true, then let's assume that q and w must meet domain of fuction of witch they are argument.

And go back to my previous question.

 

[Werg22]

This is called the Archimdean property of real numbers. You don't need to "prove" fundamental properties of the real numbers at that level, you can assume all their intuitive properties. The only exception is perhaps the least upper bound property, which isn't all that intuitive.

At a later stage, you might be interested in constructing the system of real numbers so that you can justify the consistency of the things you've been assuming. For now, don't worry about it.

 

[IB1]

Let's say that you want to prove that A=B
Then you can prove that :
A=...=...=C ,and B=...=...=C
This means that A=B.
or :
you can transform A somehow such that :
A=...=C=...=D=...=B
So basically you can transform one side, and get to the other side, or to transform both sides and to get to an obvious identity.Anyway, you 'transformation' should always be mathematically legal.

 

[sutupidmath]

I think that one of the most difficult things to teach to someone is how to prove things. Teaching someone how to solve a problem, like take a limit etc. isn't that difficult. I think that the only way to learn how to prove statements in math is to read a lot of proofs...and not just read them, but disect them, break them into small pieces and see how each of these parts plays its role in the big picture. Try to see how each assumption/hypothesis plays what role in the proof, what would happen if a certain assumption were not there, what might go wrong...and at the very end, close the book and try to do the same proof yourself. this to many people might sound like 'reinventing' the wheel, but it will be tremendously useful in the future when you face proof-type problems.

 

[hellbike]

x^n + x^(n-1) + ... + x^1 = y^n + y^(n-1) + ... + y^1

Can i assume that left and right function are equal? that seems pretty obvious, but how do i know i don't have to proof that?

 

[Edgardo]

Have a look at this thread: How to write Math Proofs
There, you'll find many introductions to proof writing and some book recommendations.

 

[wisvuze]

x^n + x^(n-1) + ... + x^1 = y^n + y^(n-1) + ... + y^1

Can i assume that left and right function are equal? that seems pretty obvious, but how do i know i don't have to proof that?

Proving something as "obvious" as that usually resorts to a matter of definition. If n is a fixed number, then the function on the 'LHS' is a function of x and the 'RHS' is a function of y. Basically, two functions are equal, by definition, if: their domains are equal, their co-domains are equal, and their mappings from each domain element to co-domain element are equal. So, assuming that these two functions are both over real numbers, then these functions are defined for all |R - and so their domains are equal. Then, you can see that the functions are the same now, since their co-domains are equal and for any element of the domain (|R) you input into both functions, you get an element of the co-domain that is common to both functions.

You can't always determine if the two functions are the same by just looking at the expression, since one function their domains could be different. For example, your function of y could've been over complex numbers while your function of x could be over real numbers. However, depending on your class, it may just be implied that everything is over real numbers.

 

[hellbike]

Still that those functions got same value for every argument need to be proved.

How to do that?

 

[wisvuze]

If proof ends with 17<18 , it this really end? Why shouldn't i proof that 17<18 is actually true?

I

Math, like any other science, is axiomatic. There will always be something at the end of the road you can't "prove". You need an initial set of statements you may take as true in order to build the rest of your system. Technically, everything you do in math in should be reduced to the original axioms of math, but depending on the subject matter, you may take previously proven statements or statements proven outside of the scope of your field for granted.

 

[hellbike]

I know that.

But i still don't know if i have to proof that

x^n + ... + x^1 = y^n + ... + y^1

for every pair of x,y where x=y
x,y in R

 

[alyks]

I highly recommend Velleman's How to Prove It: A Structured Approach. I went through the book knowing nothing about proofing beforehand and doing so helped a lot when making the transition to proofing.

 

[wisvuze]

I know that.

But i still don't know if i have to proof that

x^n + ... + x^1 = y^n + ... + y^1

for every pair of x,y where x=y
x,y in R

If x = y, then x^n - y^n = 0, also ( x^n + x^n-1 ) - (y^n + y^n-1) + ... x^1 - y^1 = 0.. et c . But it's obvious, x and y are just dummy variables for elements in the domain, if they have the same domain and the same co domain, the mapping is just F: Domain -- > CoDomain defined by the mapping of an element e in the domain to the co domain:
e^n + ... + e^1, it doesn't matter what you make that variable.

 

[Tac-Tics]

Proofs are directions to a destination. Name the major roads and landmarks. You don't have to label every side street and driveway you pass on you're way there.

The detail of your proofs depend on your audience. If you're writing a proof for differential geometry, don't waste your time with epsilon-delta proofs. Just say, "f is continuous, obviously." If you're in real analysis, though, your professor is expecting you to demonstrate you understand why f is continuous.