math assignment LUL

So like my math assignment was like

1.	a) construct a bijection between (0, 1) and (0, inf).
	b) name a bijection between (-1, 0) and (-inf, 0)

2.	a) name a bijection between (-1, 1) and (0, 1)
	b) use all of this shit to prove that (0, 1) has the same cardinality as the real numbers

So getting the actual bijective functions isn't that hard (though I got a huge hint looking at the teacher writing down f(x) = 1/x in an aside to another student after a lecture on Friday. I literally only needed that hint tbh). Proving they're actually bijections is like annoying as fuck though so like, there were different verbs so I assumed only the "construct" one needed to be proven or some shit. Anyway the functions I used were:

f: (0, 1) -> (0, inf), f(x) = 1/x - 1 (we need to sub 1 to get the (0, 1] range)
g: (-1, 0) -> (-inf, 0), g(x) = 1/x + 1 (here we add 1 to get the [-1, 0) range)
h: (-1, 1) -> (0, 1), h(x) = (x-1)/2 + 1 (slap into (-2, 0), squash into (-1, 0), bump into (0, 1) ez)

So like write out 1/x - 1 and 1/y - 1 and you can see that theyll only be the same if x = y so f is injective and shit. Okay so after that I fucking memed for some stupid reason and used f to calculate the reverse function because I hate myself (it's y = 1 - x/(x+1) and I'll give you the full shit at the bottom for NERDS) and said that this totally covers (0, 1) without really justifying it. And like then every value from (0, inf) must correspond to a value from (0, 1) because we can just run f(f^-1(y)) and get every y you know and shit. So like the function is surjective and shit, and if it's injective and surjective then it's memejec--- bijective okay. Holy shit the teacher is going to give me zero points. Anyway,

So like this means that we have (-inf, 0) covered by (-1, 0) and (0, inf) covered by (0, 1). This is like almost the entire real numbers except fuck we need a zero too. Okay let's add it then you fuck: (-1, 0) U 0 U (0, 1). Ok cool but this is just fucking (-1, 1) lol. So like (-1, 1) has the same cardinality as all the real numbers, cool beans. Okay but seriously though (-1, 1) also has the same cardinality as (0, 1) AS WE STATED EARLIER WITH OUR SUPER LEGIT BIJECTION OK so basically what it all boils down to is (0, 1) has the same cardinality as all the fucking real fucking numbers QED YOUR FACE.

So yeah that's kinda shit but hey it's like 6am fuck everything.

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Inverse function for nerds (writing f(x) is a pain so ill just use y fuck you also yes we're trying to solve for x hold your babies):
y = 1/x - 1 <=> xy = 1 - x <=> xy + x = 1 <=> x(y+1) = 1 <=> x(y+1) + y = y + 1 <=>
<=> x + y/(y+1) = 1 <=> x = 1 - y/(y+1)

We can see that like, the value range of y/(y+1) for y belongs to (0, inf) is like obviously exactly (0, 1) so that's cool. That does also mean that having 1 minus it is like not needed for surjectivity but like we wanted an inverse man. Chill.