Asaf Karagila
I don't have much choice...

Randall Munroe of the xkcd fame has a new book coming up where he explains various concepts using a small repository of "simple" words (this is based on this xkcd comic). He recently posted this blog post, where he reveals a word checker program that he wrote to help him with the task.

So I figured, why not use this for explaining mathematical theorems.

## The Challenge

Use the xkcd "simpler writer" to present a mathematical theorem. Bonus points for showing the proof as well. The only three words which are allowed outside the checker's scope are the titles "Definition", "Theorem" and "Proof". (This is a variant of "Maths, Just in Short Words" by David Roberts from last year.)

## Example: Cantor's Theorem

Theorem. Given a set A, there is no means to give each member of A a single set whose members are members of A, such that every set whose members are members of A is given to some member of A. Proof. Suppose that we have some way of giving each member of A a set whose members are members of A. Consider now the set whose members are exactly those who do not belong to the set they were given. If a member of A is given that set, then it is a member of the set it was given if and only if it does not belong to the set it was given. This is not possible, of course. So we found a set whose members are members of A, and it was not given to any member of A.

Feel free to add your simple writing of mathematical statements, their proofs and otherwise in the comments!

### There are 7 comments on this post.

By paulina
(Sep 24 2015, 23:00)

Nice! I would've gone for "way" instead of "means," but that's just me: "...there is no way to give each member..." :)

By Stefan
(Sep 24 2015, 23:34)

Definition. Let us say that A is a cool number if and only if it is a whole number at least as big as two and if one is the only whole number that can added to itself at least two times to give back A.

Theorem. For each whole number there is a bigger cool number.

Proof. Suppose not. Then there is a biggest cool number. We build a new number as follows: Let A be the smallest cool number. If there is a bigger cool number pick the least one and add it A times to itself. We again call this number A. If there is a bigger cool number than the one we picked the last time, we again add the least such A times to itself and call it A. This will be repeated until there are no bigger cool numbers to pick. Now, if the number that is one bigger than A is a cool number, we have nothing left to do. If not, then there is a cool number, call it I, that may be added to itself at least two times to be one bigger than A. By the way we built A, we know that the number I may be added some number of times to itself to be A and may then be added at least one more time to itself to be one bigger than A. But the number I is at least as big as two, because it is cool. This is not possible.

By Asaf Karagila
(Sep 25 2015, 00:06 In reply to Stefan)

Admittedly, I'm not entirely sure what is a "cool number". Can you also give the "usual" mathematical definition?

By Stefan
(Sep 25 2015, 00:13 In reply to Asaf Karagila)

Usually, we call cool numbers 'prime'. As every word describing multiplications seems to be permitted, I had to take the ugly 'add some number of times to itself'-route...

By
(Sep 25 2015, 04:52)

A whole number times itself is never two times another whole number times itself.

Here's Why: If there were such a number, there'd be a smallest such a number. If the smallest first number isn't two times anything, then times itself still isn't two times anything. If the smallest first number is two times something, then times itself is four times that something times itself. But then four times something times itself is two times another number times itself, so that other number times itself is twice that something times itself, but that other number is smaller than the first one. but we said the first one was the smallest! This can't be happening!

By Asaf Karagila
(Sep 25 2015, 04:58 In reply to Jesse C. McKeown)

Awesome. Double bonus points for "Here's Why" instead of "Proof", and triple bonus points for "This can't be happening!" :-)

By Hagen von Eitzen
(Oct 23 2015, 00:33)

(I was surprised in how nice a way one can write many things, even though some things get less easy, like: I avoided the use of the normal short one-letter names for things and sets and maps and had to call them "first nice map" and such; the use of "and so on" in one point may not be fine but I think it is still fun to read)

Theorem. If there is a map from one set to a second set and a map from the second set to the first set and both maps are one-to-one, then there is a map from the first to the second set that is both one-to-one and onto.

Proof. We look only at the case that no thing is in both given sets. The other case, that is if there are some things that are in both sets, is left for your own training - don't worry, it is quite easy.

Consider a new - third - set that has in it the things that are in the first set and the things that are in the second set (and nothing else). We have a new map, which we call the forward map, from the third set to itself: For a thing in the first set use the first map and for a thing in the second set use the second map. This idea is okay because there is no thing that is both in the first and in the second set. Note that the forward map is one-to-one (mostly because the given maps are one-to-one). We have another map, the back map, from the image of the forward map to the third set: For a thing in the image of the forward map we say that the back map maps this thing to the one and only thing in the third set that maps to it under the forward map.

Consider a thing in the third set. We try to use the back map on it, and if we can then use the back map on the image of it under the back map, and then on the image of that under the back map, and so on. After a few (or no) times we may come to a thing that is not in the image of the forward map so that we cannot use the back map any more, in which case we call this last thing the end of the given thing. Or we may be able to repeat forever, in which case we say that the given thing has no end. Now we have a nice map from the first set to the second set: If a thing from the first set has its end in the first set, use the forward map, else use the back map (which is okay, because the thing has its end in the second set or no end at all so that it must be in the image of the forward map). We also have a nice map from the second set to the first set: If a thing from the second set has its end in the first set, use the back map (which is okay to do), else use the forward map. By the way we picked these two nice maps, it is clear that starting from a thing in the first set and using the first nice map and after this using the second nice map takes us back to the starting thing It is also clear that starting from a thing in the second set and using the second nice map and after this using the first nice map takes us back to the starting thing. This shows that both nice maps are one-to-one and onto.

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