From: Stephen Dalton
Date: October 8, 2009 12:49:24 AM EDT
To: James Propp 
Subject: Comments on assignment #1 for 92.305

FIRST COMMENT

If you are trying to show something is true, you must not use its truth as a step in your argument.  To see what I mean, look at Ex. 1.2.10 from assignment #1.  The exercise wants us to use an inductive proof to show that $y_n < 4$ for all $n$.  This means, quite precisely, that given $y_1 < 4$ we need to show this exact logical implication: if $y_n < 4$, then $y_{n+1} < 4$.  The direction here is important!  The statement following the "if" is the only one we can use as a step in the proof.  Not just in this problem, but in math proofs in general, the thing following the "then" can never be written as a step in the proof.  It has to be shown to be a result of the steps in the proof.

In case none of that makes sense, here's an example.  It is tempting to write up a proof that reads like this:

(1) Show $y_1 < 4$.
(2) Assume that $y_n < 4$ (the inductive hypothesis) and state that we need to show $y_{n+1} < 4$ by necessity.
(3) Begin with the inequality $y_{n+1} < 4$ and algebraically manipulate it to obtain the inequality $y_n < 4$, which we assumed true.
(4) Claim to be finished, since we have shown the conclusion to be equivalent to the hypothesis.

To be perfectly clear, this is not an acceptable proof.  The error occurs in (3), specifically where we "begin with the inequality $y_{n+1} < 4$."  This inequality, since it is the intended conclusion of the proof (it is what follows the "then" [as opposed to what follows the "if"]), is never permissible for use in a step of the proof.  To use the intended conclusion of the proof as a step in the proof itself is a recognized logical fallacy called "affirming the consequent," and is unacceptable in proof-based mathematics.  A correct version of this same proof would have (1) and (2) unchanged, but have (3) switched in direction; we need to begin with the inequality $y_n < 4$ (since this is what follows the "if") and then algebraically manipulate THAT inequality in order to obtain $y_{n+1} < 4$ (which is what follows the "then").

Think of a proof as a recipe for a cake.  In no recipe for a cake is "cake" listed as an ingredient.  The cake is what results from correctly combining all the ingredients.

SECOND COMMENT

It is often convenient in mathematics to denote differences between objects by subscripts.  For instance, look at exercise 1.2.2(a) and (b), where the sets in question were all "A" with a subscript (1,2,3,...).  The basic reason for doing this is so that we never run out of ways to differentiate between objects; if I decided that I was going to instead denote the difference between one set in this "sequence of sets" and the next set in the same sequence by writing A for one, then B for another, then C, D, etc., then I had better hope that the sequence has fewer than 26 sets in it, or otherwise I'm going to run out of ways to distinguish between them!  So in math, in general, we write $A_1, A_2, A_3, \ldots A_n, \ldots$ (the "\ldots" is an ellipsis in LaTeX math mode) to denote a collection of sets.  Since we never run out of numbers, we never run out of ways to distinguish between them.  Most people seemed to understand this, but some students got into some hot water with how the subscripts are actually used.

It is not acceptable to say that "the intersection of all the $A_n$'s is just $A_n$."  Such a statement does not make sense.  The $n$ in this case is only meant as a sort of "stand-in" for indices of specific sets.  If the sequence is spoken of as $A_n$, then each individual set would need to be $A_1$, or $A_5$, or any other number.  It IS acceptable to write, for example, "the intersection of all the $A_n$'s equals $A_1$."  Or we could say "the intersection of all the $A_n$'s equals $A_m$ for some finite $m$," and then maybe investigate specifically which "finite $m$."  The crux here is that it is not acceptable to denote a specific set with the same letter that is used to denote the general term of the sequence.

Oh!  And also a general note about the intersection of a bunch of sets.  Most students probably understand that the intersection of a PAIR of sets is the set of all things that are "in both" - this was probably taught in your probability and statistics course, if you've taken one.  Likewise, the intersection of three sets is what's in all three, intersection of four sets is what's in all four, etc.... the intersection of an infinite collection of sets is the set of points that are in all of the sets in question.  A reasonable protest might be "ALL?  But there are infinitely many of them!"  Well, true, we can never reach the "end" of an infinite sequence of sets.  But the intersection of an infinite collection of sets $\{A_n\}$ as $n$ ranges from 1 to $\infty$ can be defined as:

{ x : x \in A_p for every integer p satisfying 1 ≤ p ≤ m and every positive integer m}.

So basically, the point $x$ is in the intersection of the entire infinite sequence of sets if the following is true: whenever we fix a finite integer (m), and take all the sets in the sequence "up until" that number (allow the integer p to range from 1 to m), the point $x$ is in all those sets (all the A_p's).  The fact that we can do this for ANY finite integer (that is, any finite m) is what assures us that this $x$ is in the intersection.  The intersection of the entire sequence consists of all points $x$ satisfying this condition.

Ok, that's all for now!

-Steve

ADDENDUM

One more thing.  Many students submitted assignments that were "not quite right" with formatting.  I strongly encourage students to make use of the template provided by Prof. Propp, or alternately, to beef up on LaTeX skills until you can produce something that looks similar.  (Prof. Propp's default template is a very standard-looking LaTeX document and it's not hard at all to get your output to look that way.)

One thing that I WILL be deducting points for in the future is failing to copy down the problems in the assignment.  This requirement is specifically stated in the opening lines to the homework template.  For each problem, you just need to take what's on the page in the textbook and make it happen in your document before you start writing up your solution.  And no, you don't need to get it to look "exactly like" what's in the book; although changing the actual problem is of course a no-no, getting something that states the problem correctly but looks a little different from the textbook is absolutely fine.

Formatting will begin to count for more later on in the semester when you should be getting more comfortable with LaTeX.

-Steve