Math Major Problems

So to show that f: A -> B is injective, we need to show that for all b in B, there is at most one a in A for which f(a)=b. I would show this by contradiction; i.e. assume f(a)=f(a')=b and see what falls apart when you assume a is different from a'. (Hint: think about what it means for something to be a function--is (gof) a function?)

To show that g: B -> A is surjective, we need to show that for all a in A, there exists some b in B such that g(b)=a. So let a be in A--can we find something in B which maps to it? (Hint: think about what you're really doing when you compose f with g; i.e. note that (gof)(a) is another way of writing g(f(a)).)

Hey, I remember you! Out of curiosity, what's the audience for your magazine? (i.e. Is it being read by just "math people," or is it supposed to be geared towards people of all interests/backgrounds?) I just think it's something to think about which would likely influence the type of content you want to include.

What you have so far sounds cool. Other things you could maybe think about including are short/simple but interesting proofs and/or ones that can be illustrated with just a picture: example. (Even though it's pretty basic, I think the simplicity of it is still super cool.) You could also do something about fractals, just like a brief description of what they are and then some pictures of interesting ones and/or ones that occur in nature. A math/number game or puzzle could be fun to add as well. These are just some ideas to maybe break up some of the info in your magazine since it seems pretty text-heavy at the moment--not sure if they follow the requirements of your assignment though.

If you need more written content though... What is it that you find fun/interesting about math? You could maybe include some information on that since it's something you're passionate and excited about, whether you frame it as purely factual or as more of a personal piece.

It’s funny because I was torn between math and physics when I was looking at schools too; I was actually leaning more towards physics until I got to college. When I went to sign up for classes during orientation, due to scheduling conflicts, I had to choose between physics and calc III. The fact that I ultimately chose calculus over the physics class (I just couldn’t not take a math class) was what made me realize that math was actually what I liked best. (Looking back, it makes sense: the thing I liked most about physics in high school was that it was basically just an extra math class.)

Now, obviously that was pretty much just random chance that I was forced to make that kind of decision so early on; although it did help me figure out my priorities pretty quickly, haha. I’d suggest looking at schools that offer both physics and math and then take a class in each when you get there to see which you like better. (Some schools even let you sit in on classes when you visit them.) And remember—you don’t even necessarily have to choose between the two; you can always double major or major in one with a minor in the other. Given that they’re fairly similar fields, there’s probably some overlap in the requirements, making a double major or major/minor less of a scheduling nightmare. Engineering could also be a field worth looking into if you’re interested particularly in the applications of math and physics.

As for what to do with either degree, your options are probably pretty numerous. My school is really big into research/academia so that’s pretty much all they push on us, but there are lots of other things out there too—I’m just not super aware of specific examples since I'm currently looking into attending grad school in a completely unrelated field. Education’s a pretty standard one if you’re interested in teaching, but I’m sure an advisor, upperclassmen, or your school’s career center could give you more examples that are specifically tailored to your interests once you get to school.

Oh man, I know exactly the type of person you're talking about--like, it must be some sort of unwritten rule that every math department has at least one of those people in it. The math department at my school is pretty small, which means I've had a lot of classes with "that guy." Like you mentioned, it did kind of ruin my appreciation of math in that I dreaded going to the classes I had with him and at one point even elected to not take a class I really wanted to just to avoid him since I knew he'd be taking it. It's gotten better over time though, especially after last semester when we had a class together with a professor who basically just refused to put up with his nonsense and would cut him off whenever he started acting like that. Other things that helped were talking with other people who felt the same way--I mean, obviously gossiping and complaining about someone isn't exactly the most mature way to handle things, but I don't know, it feels good to vent and it's an easy way to bond with your classmates ;).

Hey math majors! I know I've fallen behind on posting submissions, but in the mean time, enjoy these proof "tips."

Okay, so first you want to prove your "base case" - i.e. prove the statement for n=2. (Normally you start with n=1, but for things like sums, it makes more sense to start with n=2, as using n=1 would lead to a trivial statement and wouldn't help in the next part.) So basically you just have to show that d/dx(f_1(x)+f_2(x))=f'_1(x)+f'_2(x). (Now, I don't know what you've already covered in your class, but this is probably something you're allowed to just accept as true.)

Next is the "induction step." This means that we just assume the statement is true for n=k (i.e. d/dx(f_1(x)+...+f_k(x))=f'_1(x)+...+f'_k(x)) for some finite natural number k greater than or equal to 2 (our base case). We don't have to show this at all, we can just assume it's true and use it to show our statement still holds for n=k+1:

d/dx(f_1(x)+...+f_k(x)+f_k+1(x))=d/dx(g(x)+f_k+1(x)), where g(x)=f_1(x)+...+f_k(x) (since a sum of functions is itself a function)

from our base case, we know that the derivative of the sum of two functions is the sum of their derivatives; in our case: d/dx(g(x)+f_k+1(x))=g'(x)+f'_k+1(x)

using substitution, g'(x)+f'_k+1(x)=(d/dx(f_1(x)+...+f_k+1(x)))+f'_k+1(x)

now here's where our induction hypothesis comes into play; we can rewrite (d/dx(f_1(x)+...+f_k+1(x)))+f'_k+1(x) as (f'_1(x)+...+f'_k(x))+f'_k+1(x)=f'_1(x)+...+f'_k+1(x)

therefore d/dx(f_1(x)+...+f_k+1(x))=f'_1(x)+...+f'_k+1(x)

So now we're actually done, as this is enough to prove that the statement is true for all finite values of n. Usually you just stop after proving the base case and induction hypothesis (save for perhaps a final sentences or two wrapping up the entire proof), but for the sake of further explanation...

We've showed that this statement holds for n=2 in our base case. So what about n=3? Well, in the induction step, we showed that the statement holds for n=k+1 so long as its true for n=k. Well, let k=2 (since we know this to be true) and this implies that it's true for n=2+1=3. Now, if we let k=3 (which we now know to be true), then we know it's also true for n=3+1=4. Letting k=4 implies its true for n=4+1=5, and so on for any value of n.