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High school student here...

So recently, I have accomplished what I am considering the most exciting breakthrough of my mathematical journey so far...

I WROTE A PROOF!

I proved that the sum of any two consecutive odd numbers is always a multiple of 4

I posted my proof here and received feedback on how to improve it and where I made logic errors. I've also managed to prove that the sum of two consecutive odd numbers is always even as well (a lot easier once you get the hang of it). At this point, I don't want to stop and feel a need to prove something else - hence why I'm posting this at nearly four in the morning. Does anyone have any good ideas of something a high school student would understand and be able to prove? I tried to look for some problems online, but I keep coming across the same things over and over again.

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    $\begingroup$ A classical example of a first proof that one does in first year university, that I think is perhaps challenging enough for a high school student, would be to prove that $\sqrt{2}$ is irrational, i.e. cannot be written as a/b for two integers a,b. $\endgroup$ – Christopher.L Sep 10 '18 at 8:00
  • $\begingroup$ @Christopher.L Starting a google doc for it right now! $\endgroup$ – CaptainAmerica16 Sep 10 '18 at 8:01
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    $\begingroup$ Geometry is a marvelous field for practicing writing proofs. There are so many things you can show about circles and triangles. $\endgroup$ – Arthur Sep 10 '18 at 8:06
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    $\begingroup$ @CaptainAmerica16 Two-column is a good technique though. It is a good way to keep track of what you know and what you need for the times when the path forward isn't clear and you're just trying things. It is not so mych a type of proof (like induction, or contradiction), but more of a mental help on the go. As such, it really shouldn't be visible in the final proof, but rather in your own notes. But, just like studying techniques, school some times goes overboard on the how without really helping students understand why. $\endgroup$ – Arthur Sep 10 '18 at 8:17
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    $\begingroup$ @Christopher.L Update...I finally proved that $\sqrt{2}$ is irrational. Thanks for the problem suggestion. It was really fun! $\endgroup$ – CaptainAmerica16 Sep 25 '18 at 1:23
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At this point, I can just as well make my comment into an answer, and also elaborate a little bit on it, since I think this is good advice for any high school student.

Note: I am not familiar with what mathematics they teach at high school in different countries, so this suggestion might be too simple, but at least it shouldn't be too hard, trust me.

As I suggested, and as you managed to prove, a proof that one does in first year university, is to prove that $\sqrt{2}$ is irrational, i.e. that it cannot be written as $\dfrac{a}{b}$ for two integers $a,b$. I think this is perhaps challenging enough for a high school student.

Since you wanted a little more abstractness, I can also suggest the following:

  1. Prove that, for a function $f:X\to Y$, we always have $A\subset B\Rightarrow f(A)\subset f(B)$.
  2. Prove that, for any finite sets $A,B$, we have $(A\cup B)^c=A^c\cap B^c$, and $(A\cap B)^c=A^c\cup B^c$. Where $A^c$ is the complement of $A$

Setup and hints:

So for both of these you will need to read up on (or recap) some basic set theory, and I mean basic; just search for 'basic set theory' or something on youtube and get familiar with unions, intersections, and complements. Draw diagrams to get some intuition for things. For example, it should feel obvious after this, by actually looking at a Venn diagram, that $A\setminus B=B^c\cap A$. You may already have done all this, then all you need is to read up a little on functions and get familiar with some notation, such as the set builder notation, e.g.

$f(A)=\{y\in Y|y=f(x), \text{for some }x\in A\}$.

In plain English this would be read as: The image of the set $A$ under $f$ is equal to the set of elements $y$ in the set $Y$ for which it is true that $y$ equals $f(x)$ for some element $x$ in the set $A$. After this I think you should be ready to prove both (1) and (2) above.

I won't provide much hints in the classical sense, since these are easy proofs that you can look up (e.g. (2) is called De Morgan's Laws); But I will provide some guide lines:

Both proofs will follow a similar rather direct technique that is common when working with sets. The hard part is not going to be the steps themselves, but the abstractness of it depending on how much similar things you have done (perhaps this will be very easy for you, then hopefully someone else will enjoy them). Both can be proven in a straight forward direct manner. So, here is perhaps the only concrete hint: For (1), start out by saying "Let $x$ be an element of $f(A)$ (i.e. $x\in f(A)$), and then write down exactly what $f(A)$ is, and see what you can derive from that, and so on, always keeping in mind that you can use that $A\subset B$. For (2): If you show that $C\subset D$ and that $D\subset C$, for two sets $C,D$, then you have showed that $C=D$. And to show that $C\subset D$, you again start out with an $x\in C$ and derive in steps that then $x$ must also be in $D$.

These proofs will demonstrate very simple, but common techniques that you will have to use again and again in higher mathematics, where the outline will be very much the same, only a lot harder steps; So in that sense, they are a very good way of preparing someone for doing higher mathematics at university.

However

If these seem too simple, or you have moved on, I would actually suggest looking up group theory. It is much easier getting into than it sounds, gets fun quick, and it definitely starts to build you up for abstractness. This is usually something you do in algebra courses at university, but if you keep it simple, i.e. basically only intend to understand the definition of a group, and prove that something is a group, it will be quite easy, and train your abstract thinking. All you need to do is basically google 'easy introduction to group theory', or look up some book on an appropriate level, I found: A Friendly Introduction to Group Theory, by David Nash, this also seems to have a nice set theory section in the beginning. However, I have not read it my self, so I can't recommend it, it just looks to be on an appropriate level at first glance. Exercises on introductory group theory tend to often be proofs that sound harder than they are and are often very direct, easy, and a fun way to get into proofs and more abstract mathematics in general.

Hope this helps, and good luck!

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  • $\begingroup$ Thanks for this! I have done a few set theory problems (show a certain set in tabular form etc.), but never an actual proof or Venn diagrams, so this should be a challenging start. I have learned the definition of a group, but that was as far as I went since I figured anything else would be over my head. This was definitely encouraging :) $\endgroup$ – CaptainAmerica16 Sep 25 '18 at 12:53
  • $\begingroup$ Is ⊂ representing a subset or proper subset? It seems like it's used both ways in different books and examples. From what I know, it represents a proper subset. $\endgroup$ – CaptainAmerica16 Oct 2 '18 at 19:31
  • $\begingroup$ First of all, above, I'm using $\subset$ to mean $\subseteq$. I would say that very often, if not otherwise stated, $\subset$ just means subset (which the latex '\subset' also suggests), i.e. $A\subset B$ is the same as $A\subseteq B$. Yes, there are people that use $\subset$ to mean $\subsetneq$, and in a way this mimics the well adapted $<, \leq$ notation. But, for sets, there does not seem to be a consensus. See math.stackexchange.com/questions/50253/… for a more elaborate answer. $\endgroup$ – Christopher.L Oct 2 '18 at 20:46
  • $\begingroup$ Oh, ok! One of my biggest hurdles has been getting used to all of the notation. I'm starting to be able to read the statements without a reference. $\endgroup$ – CaptainAmerica16 Oct 2 '18 at 20:48

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