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3. Comments on Homework 1

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While grading your first homework assignment, I have noticed some alarming patterns in mathematical writing, so I thought I would share some of these with you. This way, you know what kinds of mistakes you’re making, how to spot them, and how to avoid them in the future. In addition, I’ll share my grading scale so you understand what I’m looking for. If you’re reading this on Monday, your grades may not have been published yet.

Intuitive, but not Obvious.

Lots of things are intuitively obvious and mathematically obvious. These are statements that you are allowed to assert, and some examples are:

• The identity function on a set to itself is bijective.
• There are finitely many positive integer solutions to \(n+m=5\).
• The composition of two bijective functions is bijective.

However, there is a class of statements that are intuitive but not immediately mathematically obvious, such as:

• Let \(A\) and \(B\) be sets, and suppose \(\left\lvert A \right\rvert\leq \left\lvert B \right\rvert\) and \(\left\lvert B \right\rvert\leq \left\lvert A \right\rvert\). Then \(\left\lvert A \right\rvert=\left\lvert B \right\rvert\). (This was problem 7 on the homework.)
• There is a bijection \(\prod _{n=1}^{\infty} \left\lbrace 0,\ldots, 9 \right\rbrace \to [0, 1]\). Many of you asserted this without proof, and it’s tempting to send a sequence of digits to a decimal number with exactly those digits. However, this is not bijective because decimal expansions aren’t unique.

These are very tough to tell apart, and understanding the difference is a part of learning how to do math! You should think through (though perhaps not write out) a fully detailed proof of every claim you make. This means proving things more or less from scratch, step by step! With the first set of statements, it’s straightforward enough, but with the latter set of statements, it’s far more involved.

If you make an intuitive but mathematically nonobvious claim, you don’t necessarily have to spell it out in full detail. However, a complete proof necessitates some level of justification rather than a raw assertion.

A common mistake that was made along these lines was justifying that the countable union of countable sets is countable. Many of you used the diagonal argument and asserted that this was in bijection with \(\mathbb{N}\). You may not want to exhibit this bijection and prove that it’s a bijection (though some of you did). This “zig-zagging path” certainly produces an injection (an example of a mathematically obvious claim), and it’s a surjection since every element is reached within finitely many steps (mathematically less obvious).

As such, exercise extreme caution when saying things like “clearly”, “obviously”, “observe”, “notice”, etc. You are allowed to use them, but I found numerous errors that were preceeded by one or more of these terms.

Complete Proofs vs. Detailed Proofs

While I don’t expect a fully detailed proof for each problem, I am expecting a complete proof. A fully detailed proof might prove every single claim step by step; this might be a bit too much. For instance, you do not need to prove that the composition of bijective functions is bijective.

A complete proof is one that writes a justification for each step of the argument, but does not necessarily prove each of these justifications. Rather, it proves the “critical components” of the proof and might leave out some routine or obvious details. For instance, if you compose two bijections and utilise the fact that the composite is bijective, then you should at least mention the fact that it is bijective. Likewise, if you use the fact that an injective set function is actually a bijection when you restrict its codomain, you should state this fact, even if you don’t prove it!

I understand that this point is somewhat hazy and vague, and unfortunately there’s not really a precise or universal way to describe what you should prove and what you can leave out. When in doubt, you can always ask.

Notation

If you are using a mathematical symbol or a variable, you need to explicitly say what it means! I cannot deduce what \(A\sim B\) means when \(A\) and \(B\) are arbitrary sets. You can certainly reuse notation from a different problem, but you should still explain that you are doing so: “Let \(A\) and \(B\) be the sets from problem 14…”

In addition, some notation can be very misleading or prone to error; I saw a couple of proofs that had the right idea but said something incorrect because of poor or ambiguous choices of notation. This is particularly problematic when you use the same letter to mean different things throughout the proof, especially if you use the same letter twice at the same time. Most commonly, these notational issues arise in indices for unions, sums, and sequences, especially when multiple of these indices are in use at the same time.

Finally, please stick with common notation. You are more than welcome to redefine notation to your own preferences, but please do not say things like “let \(\mathbb N\) be the set of real numbers”.

Citing Results

A handful of proofs cited well-known facts, particularly Cantor’s diagonal argument in proving the countable product of countable sets may be uncountable. While it’s great that you are aware of this argument, it’s not enough to cite the arugment as your proof. You are still expected to write a complete proof! If you’re not sure if you can use a well-known theorem or argument, please ask me!

Diagrams

For problems like the countable union of countable sets being countable, quite a few of you provided diagrams to illustrate your proof. This is very good and it helps clarify your argument, but keep in mind that you still need to prove everything you are saying. Proofs by pictures — that is, citing your own diagram and asserting its validity — do not constitute complete proofs.

Moreover, while a picture can depict or describe a construction, function, etc., you are still expected to write something down for it. It’s not a great idea to say, “Let \(f\) be the function drawn above,” you still need to write down in words or in equations what \(f\) actually is.

Grading Scale

I tried being very lenient with grading, and the scale is more or less as follows:

• 0/10 — No submission.
• 5/10 — An incorrect proof that does not give the right arguments or constructions.
• 7/10 — A proof that generally uses the right arguments and constructions, but has major flaws or omissions in its justifications. Errors of this sort are mathematically incorrect statements and mistakes; omissions are of details that are crucial to the mathematical validity of the argument.
• 9/10 — A proof that is nearly correct, but exhibits subtle errors or omissions in reasoning. This could be asserting something that is true, but not realising that you are implicitly using an assumption in the problem in doing so.
• 10/10 — A correct proof.

Some errors are small and inconsequential, such as mislabeling an index in a sum or a notational error. You received -0.05 points for these if that’s the only mistake you made. This is so you click on the big red 9.95 and read the feedback. Some question-dependent adjustments may be made too.

Update: After conversing with Professor Greene, we have decided that this scale is too lenient. Sorry! Instead, you will get 0/10, 2/10, 5/10, 8/10, and 10/10 for each “step” of the scale, with possibly \(\pm 1\) or \(\pm 2\) point deviations for errors and omissions. This will be in effect starting homework 2 and beyond. In any case, just focus on good reasoning and good proofwriting.