/sci/ - Science & Math

well i'm off to bed, but i'll just go ahead and say
if you've defined the numbers like i did here >>7995958
and assigned the symbol "1" to the element [math] \{ \emptyset \} [/math]
then 1 = 1 follows directly from the axiom of extensionality

[math] \forall{A}. \forall{B}. (\forall{C}.(C \in A \iff C \in B) \implies A=B) [/math]

which says "if you take any sets A and B, and if for every set C, C is in A only if C is in B, then A is equal to B"
which, put less autistically, says "if A and B have the same elements, then they are equal"

so there's my first proof that 1 = 1

i get bored in class and do a lot of messing around with foundations and low-level stuff so i've got a lot more of this if anybody's interested

i kinda want to post it all TЬH but i'd rather post it in a thread where there's interest
i have category theoretic and type theoretic proofs that 1=1 as well, but they're pretty much the same as that proof up there^

also can somebody tell me where pic related is from?
i've image searched it and stuff but i can't find anything
i just want to know why the dodecahedron group is off by itself as if it's as important or even as interesting as

Do you guys have cool diagrams that sum up different areas of maths into one neat space ?

pic related : example

So I saw this (attached picture) relationship on another thread .

Does anyone else have a similar illustration, which shows the relationship between different branches of maths ?

>Mathematics

In the meantime, I will self-bump with good visuals and links that have helped me, for any /sci/ browsers out there that teach themselves things.

>Analysis
https://59clc.files.wordpress.com/2012/08/real-and-functional-analysis-lang.pdf

http://classicalrealanalysis.info/com/documents/TBB-AllChapters-Landscape.pdf

>Engineering Textbooks
http://kisi.deu.edu.tr//ali.sevimlican/Peter_V__O%27Neil-Advanced_Engineering_Mathematics,_7th_Edition__-Cengage%282011%29.pdf