zooxanthellae said:
I'm looking at this guide right now. I'll be heading here in the Fall, so I figure this style is something I should get used to. Thoughts?
The guide has some good advice on the errors beginners often make, however the style of proof it uses is NOT standard. I suspect it uses this format for there to be no ambiguity, however in real math proofs we use English, but we use it carefully. "For all real x there exists a unique integer n such that n\leq x < n+1." is a good way to state what could also be written:
\forall x \in\mathbb{R}\,\exists n \in \mathbb{Z}\, \forall m \in \mathbb{Z} \left[m \leq x < m+1 \Leftrightarrow n=m \right]
if we insist on avoiding English. The guide seems to settle on some kind of middle-ground, but real proofs use less logical symbolism. One reason to use this much logical symbolism is to force yourself to think in a logically meaningful manner.
Try taking a look at one of the proofs in Spivak. That is how a real proof looks. Universities do not have their own "proof notation". Everyone has their own style and people can read other people's style of proof.
To give an example of the kind of proof I would expect from someone who has completed their a course doing proofs (something very similar to this is probably an exercise in Spivak):
Theorem (AM-GM inequality in two variables): For all non-negative real numbers x and y we have
\frac{x+y}{2} \geq \sqrt{xy}
with equality if and only if x=y.
Proof:
Let x and y be distinct non-negative real numbers. Then \sqrt{x} \not= \sqrt{y} so \sqrt{x} - \sqrt{y} \not= 0. This implies (\sqrt{x}-\sqrt{y})^2 > 0. Expanding using the rule (a-b)^2 = a^2 + b^2 - 2ab we get:
0 < \sqrt{x}^2 + \sqrt{y}^2 - 2\sqrt{x}\sqrt{y} = x + y - 2\sqrt{xy}
Rearranging the terms and dividing by 2 we get:
\frac{x+y}{2} > \sqrt{xy}
We have now proven that if x=y is false, then the inequality holds, but without equality. It now only remains to show that if x=y, then:
\frac{x+y}{2} = \sqrt{xy}
But this is immediate by the following calculation:
\frac{x+y}{2} = \frac{x+x}{2} = x = \sqrt{xx} = \sqrt{xy}
Note that at several points I just assumed results without referencing them since these are basic (what is basic of course depends on your level and when you write expository work: your audience's level). For instance I wrote \sqrt{x}, but does x actually have a square root? It does because I assumed that x is non-negative. If you were a bit unsure about this, then there would be nothing wrong with adding:
Since x and y are non-negative they both have unique square roots.
I also assumed that since x and y are distinct we know that \sqrt{x} and \sqrt{y} are distinct. This is well-known, but you cannot simply accept it on faith. There are plenty of other functions for which it does not hold in general. For instance if a and b are distinct real numbers I can't deduce that a^2 and b^2 are distinct since we may for instance have a=1 and b=-1. I have used plenty of other small facts that needs proofs, but I assume they have already been given (I use \sqrt{x}^2 =x and \sqrt{x}\sqrt{y} =\sqrt{xy})
The level of detail, and what details you choose to focus on depends on your level. At your level it would probably be prudent to add notes such as "it's well-known that if x and y are distinct, then so are their square roots." I believe Spivak actually proves this, and until that point you can't reference it. As you evolve as a mathematician so will your proof style. If the same proof were given in a graduate class it would probably just read:
"Follows trivially from (x-y)^2 \geq 0 with equality if and only if x=y."
or more likely:
"Exercise for the reader" or "Trivial".
because at this level people have so much experience that they can easily reconstruct the proof, probably even just do it in their head. If there is any step where you are not sure exactly why the next part follows, then you have not provided enough details. The authors of your books may get away with shorter proofs than you, but that's because they are more experienced and can easily see how the proof works even if you leave out a couple of simple steps. Also a gap in a proof in a book is often taken as an extra exercise.
You may be at a level where you can't really see when we use commutativity, associativity or distributivity, and in that case you should add "By associativity we see ..." and statements like that, but at some point this should become trivial for you to work out in your head so you may write:
Using the definition of squaring and the usual arithmetic rules we get:
(a+b)^2 = (a+b)(a+b) = aa + ab + ba + bb = a^2 + 2ab + b^2
Once such manipulations become simple you may simply write:
(a+b)^2 = a^2 + 2ab+b^2
since it's obvious that to check it we just multiply out and collect terms.
Last quarter I was enrolled in an algebra course using the text Algebra by Hungerford, and our professor commented that Hungerford often follows up a theorem with what he calls a "proof sketch", but despite the convention to call these arguments proof sketches we should consider them proofs. Basically Hungerford targeted an audience a little less experienced than us so his "proof sketches" had gaps large enough that he felt that the reader should fill in the details, however a somewhat more experienced reader would easily be able to fill in the gaps and consider the proof sketches perfectly acceptable proofs.
Another thing I would like to note is that a proof is an expository work meant for other humans to read. Therefore there is no shame in helping them a bit. If your proof becomes a bit cluttered try to clean it up, and up sentences explaining what you are intending.
1) "We shall start by proving (2)"
2) "Let us first start by proving the statement with the extra assumption that n is a prime number".
are both perfectly acceptable parts of a proof.
Anyway I think I have rambled enough, just take a look at the proofs by Spivak to get a feel for what proofs are. Hope this makes sense.
