Couldn’t get this idea outta my head, whether it’s reasonable or not…

Happy Thanksgiving!
— 

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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6 thoughts on “Couldn’t get this idea outta my head, whether it’s reasonable or not…”

    1. 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.

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