There are so many things wrong with this problem that it made me make a blog post about it. The problem, of course, is that it relies on several conventions that are taught in math classes, but are not the conventions when you actually do math (or at least they aren't the conventions in every field of math).
First of all, I strongly object to the use of the word ``domain'' in this context. The domain of a function is absolutely not dependent on the definition of the function. A function is defined with a specified domain and codomain, of which this problem specifies neither. Instead, it tries to implicitly define the domain from the properties of the function. This is commonly used in math classes. I know I learned in some math class ``how to find the domain of a function'' such as
, but I have never seen this outside of math class and a few competitions (and all of the competitions that I've seen it on, including the AMC, are very clearly tailored for average math class students, or at least students who don't have math education beyond that which you get in the classroom). Nevertheless, while I object to the use of the word ``domain'', it was clear what the AMC meant, so that would be admissible.However, the real problem comes in the use of
, which is clearly and unambiguously defined as 
. Furthermore, 
has a well-defined value. The problem is that 
has different meanings in different fields, and there is no way to know which one the AMC wants, except for the fact that people who have not learned math outside of the classroom can only be expected to know one of them.The AMC never specfiied a codomain. And actually, since the AMC assumes the knowledge of complex numbers, this is a huge problem.
is, for a vast number of fields, given the value of 
, even though any one of 
would work just as well. However, the point is that it is defined.If I were to ask someone what is the domain of
, I would almost certainly get the answer 
. But then, what if I say, ``Oh but 
is defined as 
!''? Then the person I'm talking to will, in many cases, revise their answer to all of 
. The exact same problem exists with . Is the domain 
or 
? That question comes directly from the question as to whether the codomain is or 
.So please, if you want to use this problem on a contest, word it like this:
Well but then you have a problem with "largest set," because there are infinitely many uncountable sets satisfying that condition. But maybe I'm just being nitpicky.
ReplyDeleteThe largest set is well defined because it contains all the other such sets. It's actually not an issue at all.
ReplyDeletethat anyone besides you would be confused by this is dubious =P
ReplyDeleteI actually was not confused by this during the test, I just forgot to think about negative reals. That's why I'm not submitting this as an actual complaint to the AMC committee. But I still hate seeing this on a contest.
ReplyDeleteBrian - sorry to have to resort to contacting you via a comment on your blog but you seem to be pretty stealth otherwise and I couldn't find another way to get through to you.
ReplyDeleteI'm one of the co-founders of a Falls Church based software company called FoundationDB. We're building a new database product that, to sum its main differentiating points, is massively scalable (designed to run on clusters of 100s or 1,000s of machines), extremely fault tolerant (machines / racks of machines can blow up, it keeps humming), and keeps a data model with ACID (Atomicity, Consistency, Isolation, Durability) transaction guarantees - something that other databases addressing the first two issues have given up on. We're really excited about the project.
Why am I writing to you? Well, first off, 100% of our development team at the moment is made up of TJ grads :) Granted, we're about 10-12 years older than you ('98 and '00), but we respect TJ around here. We've been thinking it might be fun to have a smart intern this summer, during breaks, etc. but we're working on something extremely difficult and would need someone that loves hard problems and can keep up. Someone like us when we were in TJ came to mind... we did a little searching around on the web, saw some of your accomplishments, and thought we'd message you. If you're interested in learning a bit more, send me an email at nick.lavezzo at foundationdb dot com. Or feel free to internet stalk me and post a comment on my blog instead :)