Happy Thanksgiving!

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Given some binary tree (doesn’t *have* to be just a binary tree though, I guess):

A

/ \

B C

/ \ / \

D E F G

It seems possible to represent the tree with a bunch of ordered n-tuples and relations.

First, let A = (A, A). In this context, "=" really just stands for "R" (relation). And it’s a little pedantic to say that A R {A and A}, but for me it made it clear that A=(**A**, *A*) means A R **A** and A R *A**.*

With that being said, let B = (D, E) and let C = (F, G).

A is related to B as well as C, written as A = B and A = C, which could be written equivalently as (A, A)=(B, C); A R {B and C}.

But since A = (A, A) = (B, C) and B=(D, E); C=(F,G), its also true that–

(B, C) = ((D, E), (F, G)).

So, A = (A, A) = (B, C) = ((D, E), (F, G)).

**(1)**
A is related to itself and is related to B and C too. And, B is related to (D, E), and C is related to (F, G)..

Breaking **(1)** down further:

*A = (A, A):*__ __A R A ^ A = A R A = (A, A) (derp);

__A = (B, C)__: A = A R B ^ C = (A R B and A R C) = {**(A, B), (A, C****)**};

__B = (D, E):__ (B, C) = B R D ^ E = (B R D and B R E) = {**(B, D), (B, E)**};

__C = (F, G):__ C R F ^ G = C R F and C R G = {**(C, F), (C, G)**}.

Lol, I don’t know if I got some terminology right (notably relations), but its nice to know that if we look at the levels of the binary tree, taking the cartesian product of lvl1 x lvl2, then lvl2 x lvl3, we have:

lvl1: {A}

lvl2: {B, C}

lvl3: {D, E, F, G}.

Then, taking the Cartesian product of the levels:

lvl1 x lvl2 = {A} x {B, C} = {**(A, B), (A, C)**}

lvl2 x lvl3 = {B, C} x {D, E, F, G} = {**(B, D)**, **(B, E)**, (B, F), (B, G), (C, D), (C, E), **(C, F)**, **(C, G)**}.

I guess I’m not *completely* off, but the whole A = (A, A) thing is still silly and may lead to trouble though. Seems that way anyways.

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o_o

Food goes here! *points to open maw*

Turkey day was fun–Had 3 servings over the entire day and fell asleep every time xD.

I… feel stupid now. I really have no idea.

Wasn’t the intention though :<

For the most part, it was an idea I couldn’t get out of my head, yet it’s something I barely understand right now.

I’ll take another look at this and let you know what I think once my brain is unscrambled and/or beered up.

That’d be awesome.

Was recently introduced to this proof writing class to fulfill the applied math minor I really want to get.

While the class opened up a lot of ideas, I’m still pretty new to the stuff. Chances are, I’ve gotten some terminology wrong and some parts should be re-written.