Timothy Gowers, the mathematician who compiled the incredible Princeton Companion to Mathematics, has begun a series of blog posts directed toward students who aspire to become mathematicians. It’s really quite amazing! Here we have one of the foremost mathematicians of our time explaining the nuts and bolts of mathematics. One of his goals is “to stress the points that you need to understand in order to be able to write proofs.”
The thread begins with his introduction, and then proceeds with a series of posts on basic logic. His latest post is about injections and surjections, which are fundamental concepts that describe functions.
I urge all students interested in math to read these posts, no matter what level you are. While the posts are directed at college math majors, if you are in middle school and like math, it is worth taking a look at them. If you have a great deal of trouble understanding them, don’t get discouraged. The mind has a mysterious, magical way of ruminating on things at a subconscious level. By trying to understand his posts, you will be setting the stage for future advancement even if you feel like you’re not understanding a thing.
By the way, if you have already learned about the concept of the domain of a function at school (in the US), you might be confused by Prof. Gowers’s first point in his latest post on injections and surjections. That’s because, rather unfortunately, the way the term “domain” is used in the US K12 arena is different from the mathematician’s usage of the term.
When a mathematician defines a function, specification of the domain is a part of the definition. However, in the US K12 arena, “domain” is usually understood to mean the set of all real numbers for which the given function rule makes sense. This is unfortunate for a number of reasons, one of which is the implied assumption that all functions are supposed to be defined on some subset of the real numbers, which is quite limiting. For mathematicians, the function concept is far more general; any set can serve as the domain of a function.
But in the US K12 arena, you will sometimes see a math problem like this:
“What is the domain of the function ?”
The expected answer is: The set of real numbers other than zero. That’s because the function rule has meaning for all real numbers except for
For mathematicians, one is supposed to specify the domain when defining the function and it doesn’t make sense to give a function rule and ask, “What’s the domain?” In fact, in his post, Prof. Gowers gives two functions with the same function rule but different domains.
I don’t know how this discrepancy in the definition of domain came about and I don’t know if it will ever go away, though I would like to see math teachers adopt the mathematician’s definition of domain. It’s hard to imagine this happening any time soon though as the US K12 definition of domain is so entrenched you even find SAT problems like this:
“What is the range of the function defined by ?”
To a mathematician, this question cannot be answered. A mathematician wouldn’t even be able to determine the image (which is the set of all elements in the range that are, in fact, mapped to by the function) of this function until the domain is specified. But in the US K12 arena, it is supposed to be understood that the domain is taken as the set of real numbers for which this function rule makes sense and the range is taken to mean what mathematicians call the image of the function. So, in the US K12 arena, the domain of the function is intended to be all nonzero real numbers and the range is the set of all real numbers other than 2. (This question is question #30 on page 19 of the College Board publication “Getting Ready for the SAT Subject Tests” that was available on October 12, 2011.)
If you are a US student, keep this in mind when you read Prof. Gowers’s blog!