Noob proof question.

[Vastor]

1, Prove the following:

 

(i) [latex]if, ax = a[/latex] for some number [latex]a \neq 0, then, x = 1.[/latex]

(ii) [latex]x^2 - y^2 = (x-y)(x+y)[/latex]

My Answer:
(i) if [latex]ax = a, a \neq 0[/latex]

[latex]ax * a^{-1} = a * a^{-1}[/latex]

[latex]x * a * a^{-1} = a * a^{-1} = 1[/latex]

[latex]x * 1 = x = 1[/latex]

 

(ii) let [latex]y = x +k[/latex]

then [latex]x^2 - y^2 = x^2 - (x+k)^2[/latex]

[latex]= x^2 - (x^2 + 2xk + k^2)[/latex]

[latex]= x^2 - x^2 -2xk - k^2[/latex]

[latex]= -2xk -k^ = -k(2x+k)[/latex]

 

then [latex](x-y)(x+y) = (x - (x+k))(x + x +k)[/latex]

[latex]= (x-x-k) (2x +k)[/latex]

[latex]= (0-k)(2x+k) = -k(2x +k)[/latex]

 

Thus, [latex]x^2 - y^2 = (x-y)(x+y)[/latex]

My Question:

- Is my proof right & accurate?

- How to tell either the proof that I make right or no? (I can't afford to ask here every time I make one)

 

More Detail on my question:-

on first Exercise, I found it to be very obvious but I still have some doubt (which is probably unnecessary) on the proof.

 

On second one, I found it to be very obvious too if I'm going from [latex](x+y)(x-y) => x^2 - y^2[/latex] but not its reverse!

It give me headache on how to proof the existence of "xy - xy = 0" in the calculation. So, I try a different approach and get the above proof but is it appropriate? Does it doesn't matter how people proof it as long as you can put the "flow-in-between", then consider the proof as accurate?

By multiple afford searching some clue on google, I end up become more confused as I see proof got it's own group and it's specific method on how to do. Yeah, it's pretty understandable with their given example yet for meI found it very hard to apply it on my very own question.

 

Some note:

- Those above are my first two proof problems.

- Problem taken from Spivak Calculus, if anyone found better book for me, please recommend!

 

[John]

1, Prove the following:

 

(i) [latex]if, ax = a[/latex] for some number [latex]a \neq 0, then, x = 1.[/latex]

 

(ii) [latex]x^2 - y^2 = (x-y)(x+y)[/latex]

 

My Answer:

(i) if [latex]ax = a, a \neq 0[/latex]

[latex]ax * a^{-1} = a * a^{-1}[/latex]

[latex]x * a * a^{-1} = a * a^{-1} = 1[/latex]

[latex]x * 1 = x = 1[/latex]

 

This looks fine to me.

 

(ii) let [latex]y = x +k[/latex]

then [latex]x^2 - y^2 = x^2 - (x+k)^2[/latex]

[latex]= x^2 - (x^2 + 2xk + k^2)[/latex]

[latex]= x^2 - x^2 -2xk - k^2[/latex]

[latex]= -2xk -k^ = -k(2x+k)[/latex]

 

then [latex](x-y)(x+y) = (x - (x+k))(x + x +k)[/latex]

[latex]= (x-x-k) (2x +k)[/latex]

[latex]= (0-k)(2x+k) = -k(2x +k)[/latex]

 

Thus, [latex]x^2 - y^2 = (x-y)(x+y)[/latex]

 

This is also correct, but it can be shortened by a couple of lines. Notice that since [math]y = x + k[/math], then [math]k = y - x[/math]. Thus, you can substitute [math]y - x[/math] for [math]k[/math] in the fifth line of your proof to arrive at your conclusion more quickly.

 

 

on first Exercise, I found it to be very obvious but I still have some doubt (which is probably unnecessary) on the proof.

 

Sometimes the proofs that give a person that most trouble (especially those new to proof-writing) are the simplest. Your proof is correct.

 

On second one, I found it to be very obvious too if I'm going from [latex](x+y)(x-y) => x^2 - y^2[/latex] but not its reverse!

 

Yes, the reverse is a bit more involved, but as mentioned earlier, making the appropriate substitution simplifies things a bit.

 

It give me headache on how to proof the existence of "xy - xy = 0" in the calculation. So, I try a different approach and get the above proof but is it appropriate? Does it doesn't matter how people proof it as long as you can put the "flow-in-between", then consider the proof as accurate?

 

Assuming I'm understanding you correctly, don't worry. Your proof is fine, and yes, while "elegance" is appreciated in a proof, a proof is fine so long as each step is justified.

 

 

By multiple afford searching some clue on google, I end up become more confused as I see proof got it's own group and it's specific method on how to do. Yeah, it's pretty understandable with their given example yet for meI found it very hard to apply it on my very own question.

 

The basic notions of what a proper proof entails can be taught, but being good at actually seeing and writing proofs takes time and practice. Keep at it, and you'll probably find the process gets easier.

 

Some note:

- Those above are my first two proof problems.

- Problem taken from Spivak Calculus, if anyone found better book for me, please recommend!

 

Good job. Spivak is a well-written book and very thorough. I don't know if there are really "better" books, but the other two usual recommendations are Tom Apostol and Richard Courant. While Spivak's Calculus covers only single-variable, Apostol and Courant's calculus series both go through multivariable as well. Apostol is a bit more terse and technical than Spivak, Courant is perhaps more focused on applications. All three are, by all accounts, excellent. And of course, while Spivak, Apostol, and Courant are the ones I've seen recommended the most, there are countless other calculus books. If Spivak works well for you, then great. If not, then it's just a matter of trying to find a text that does.

 

Best of luck to you in your studies.

[Vastor]

In general, what I mean about the "flow-in-between" is how much for a proof is sufficient to be a proof?

let's give a look at this question:

1. Prove the following:

(iv) [latex]x^3 - y^3 = (x - y)(x^2 + xy + y^2)[/latex]

(v) [latex]x^n - y^n = (x-y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1})[/latex]

My Answer:

(iv) [latex]x^3 - y ^3 = (x-y)(x^2 + xy + y^2)[/latex]

[latex]= x(x^2 + xy + y^2) - y(x^2 + xy + y^2)[/latex]

[latex]= x^3 + x^2 + xy^2 - x^2y - xy^2 - y^3)[/latex]

[latex]= x^3 - y^3[/latex]

(v) This gonna be a bit of mess, considering that I don't really know what I'm doing.

[latex]x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1})[/latex]

[latex]x^5 - y^5 = (x - y)(x^{4} + x^{3}y + x^2y^2 + xy^{3} + y^{4})[/latex] if n = 5

[latex]x^3 - y^3 = (x - y)(x^{2} + xy + y^{2})[/latex] if n =3

As we can see, certain pattern developed here based on the first equation. Thought so, I have a bit of problem managing (+ ... +) in the original equation. So I tried this.

when n = 3, now let [latex]K = xy[/latex]

[latex]x^3 - y^3 = (x-y)(x^2 + K + y^2)[/latex]

[latex]= x(x^2 + K + y^2) - y(x^2 + K + y^2)[/latex]

[latex]= x^3 + xK + xy^2 - x^2y - yK - y^3 = x^3 + y^3 + 0[/latex]

let [latex]M = xK - yK + xy^2 - yx^2 = 0[/latex]

when n = n/unknown, now let [latex]K = x^{n-2}y + ... + xy^{x - 2}[/latex]

[latex]x^n - y^n = (x-y)(x^{n-1} + K + y^{n-1})[/latex]

[latex]= x(x^{n-1} + K + y^{n-1}) - y(x^{n-1} + K + y^{n-1})[/latex]

[latex]= x^n + xK + xy^{n-1} - x^{n-1}y - yK - y^n = x^n + y^n + 0[/latex]

let [latex]M = xK - yK + xy^{n-1} - yx^{n-1} = 0[/latex]

when n =5

[latex]M = xK - yK + xy^4 - yx^4 = 0[/latex] is supposed to be.

[latex]x^5 - y^5 = (x-y)(x^4 + x^3y + x^2y^2 + xy^3 + y^4)[/latex]

[latex]thus K = x^3y + x^2y^2 + xy^3[/latex]

[latex]x^5 - y^5 = (x-y)(x^4 + K + y^4)[/latex]

[latex]= x^5 + xk - yk + xy^4 - x^4y - y^5[/latex]

[latex]= x^5 - y^5 + M = x^5 - y^5[/latex]

where [latex]M = xK - yK + xy^4 - yx^4 = 0[/latex].

My Question:

(iv) I found another way to prove this question, by assigning random number to the variable on both equation. The equation would be true if both equation produce same result and false otherwise.

Thus, which one is better(more sufficient) as the proof for this question? or (based on your explanation, if I understand correctly), it's doesn't matter either way as long as others can read the "flow-in-between" the proof?

If so, how about question (v), is the proof sufficient enough? (I got the hard feeling it's not)

Well, at least it's because I try to mimic from this site on how to proof the series part:-

http://www.purplemath.com/modules/inductn3.htm

But, (unfortunately?) the above solution is the best I can think of, even though I realised & noted several problem that I faced:

- based on the site, There is only 1 unknown/variable which make solving the equation much simpler, should I make y = 2x or insert any random number for each x and y? I did, and I not found much or any pattern to manipulate, but end up with a mess of number.

- secondly, I don't know how to manage the "series's symbol" = (+ ... +). Actually, I had a bad time with solving +(...)*(x) - (...)*(y), which end up something un-solvable.

- based on second point, I'm not even sure if putting (K = blabla + ... + blabla) on the above solution is "legal" in the first place! Yet, I used it because the site used an equation to represent its series, and I'm sure a variable can represent an equation. Thus, variable represent a series!

- based on the site also, I tried to apply the induction process by providing a variable for (n) to represent "anywhere" and (n+1) to represent the next place or "everywhere". Again, I'm facing a big problem on managing the (+ ... +) and created another mess of equation.

Anyway, thank you for reply beforehand, really appreciated.

[John]

Your proof for Problem IV is almost fine. You should remove "[math]x^3-y^3 =[/math]" from the first line, as [math]x^3-y^3 = (x - y)(x^2 + xy + y^2)[/math] is what you're trying to prove in the first place.

 

You may be overthinking Problem V just a bit. If you look at your proof for Problem IV, what you've done is show [math](x - y)(x^2 + xy + y^2) = x^3-y^3[/math], which also proves [math]x^3-y^3 = (x - y)(x^2 + xy + y^2)[/math]. The same strategy can be used to solve Problem V. Don't be thrown off by the "[math]+...+[/math]". Just multiply the terms you see, and you may notice something about which terms cancel in the result. From there, all that's left is to simplify, and you're done.

 

Keep in mind that plugging in specific values isn't sufficient to prove an idea. You might try three values for [math]n[/math] or 100,000,000,000 values for [math]n[/math], and even if they all check out, that isn't enough to prove the statement. (Edit: This statement has obvious exceptions, so take it with a grain of salt. For instance, if I asked you to prove that some property holds for all natural numbers [math]n[/math] less than or equal to 100,000,000,000, then you could just go through 1 to 100,000,000,000, trying each one, to see whether the property holds.)

 

To answer your more general question about proofs, knowing what steps are necessary comes down to practice and knowing one's audience. For instance, you can say something like, "Since [math]x^2[/math] is even, then [math]x[/math] must be even," without having to prove that, unless you think your reader isn't familiar with that result and doesn't know how to prove it himself.

 

Even for a more mathematically experienced audience, though, sometimes more detail is preferable to less. For instance, an expert in complex analysis might not be aware of certain advanced or recent results in number theory. If you use the number theory result in some proof, it might be a good idea to at least specify (either inline or in a footnote) which lemma, theorem or whatever, you're using.

 

It can be difficult to strike a proper balance between conciseness and clarity. If you want to see many examples of how proofs can be presented, you might enjoy browsing ProofWiki (if you're not aware of it already).

Edited June 8, 2013 by John