a lesson learned too late

On Facebook, recently, I saw some discussion about a couple of professors I knew when I was in grad school at the University of Michigan. One of the professors discussed was M.S. Ramanujan, whom I had had. In that thread, I mentioned that I had my own Ramanujan story, but that I would post it on my blog. So, here it is.

But before I start, let me note that my father had taken a class with ramanujan when he was an undergrad at the UM. That's neither here nor there.

So, it was my first semester as a grad student, and I was taking a class in real analysis with Ramanujan. On the midterm, we were asked to prove disprove the following:

The set A union B is measurable if and only if both A and B are measurable.

We were allowed to assert without proof anything that had been proven in class.

My solution was simple: Let A be a nonmeasurable set of reals. Let B be A's complement. A union B is the set of reals, which is measurable. But neither A nor B is measurable. Therefore the statement is false. QED.

I only got half credit for my solution, which really angered me.

Professor Ramanujan argued that I got half credit because I only answered half the problem. Despite presenting the problem as one statement, he had intended it to be interpreted as two statements:

The set A union B is measurable if both A and B are measurable.

The set A union B is measurable only if both A and B are measurable.

I had proven that the second statement was false, but had not said anything about the first statement. Therefore I only got half credit. Ironically, the first statement is much simpler to handle, since we had proven it in class.

I argued with Professor Ramanujan that I should get full credit; he had presented the proposition as one statement and asked us to prove it or disprove it. I did so. His response was that, if I want to be a hardass about it, he's sure he could review my paper and find points to take off elsewhere. I dropped the argument.

Looking back, I have mixed feelings about it. I was certainly right in at least one interpretation of events. Of course, as one of my classmates pointed out to me, in the name of elegance, I should have written into my answer something along the lines of "Of course, as proven in class, if A and B are both measurable, then their union is measurable."

But the bigger point that I didn't understand is that it didn't really matter. In high school and in undergrad, grades on tests were crucially important, as the final grade would be some well-defined average of scores and tests and homeworks. In grad school, the professors had much wider latitude to assign grades based on how well they felt the student knew the material. So the extra points for restating such a trivial result didn't really matter.

Live and learn.