[Category theory] Basic Isomorphism Proof Checks

2021.10.18 02:16 twofingerjump [Category theory] Basic Isomorphism Proof Checks

I would like proofs checks on two theorems
(1) Prove for any set A is isomorphic to itself
For there to be an isomorphism between sets X and Y we need functions
f: X -> Y and g: Y -> X
that satisfy the following compositions:
(g o f) = id_X
and
(f o g) = id_Y
we will check these compositions are satisfied.
let g=id_A: A -> A
we see that by substitution
(id_A o id_A) = id_A
let f=id_A: A -> A
and we see that by substitution
(id_A o id_A) = id_A
thus the compositions for a isomorphism between f:A->A are satisified and hence A is isomorphic to itself
(2) Prove for any sets A, B id A is isomorphic to B, then B is isomorphic to A
Proof:
Assume A is isomorphic to B then there are functions
f: A->B
g: B->A
Such that
(g o f) = id_A
(f o g) = id_B
B is isomorphic to A if we have the compositions:
(f o g) = id_B
(g o f) = id_A
But due to A being isomorphic to B, we already have such compositions satisfied
Therefore B is isomorphic to A
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2021.10.18 02:16 TheBinarySon [SPOILERS] Leviathan questions

Playing ME3/LE for the first time (never played me3 before) and just finished going through Leviathan part. I know vaguely what happens at the end/what the choices are, but for now I have some questions based on the explanation of things given by the Leviathan:
1.) Why didn’t the leviathans just kill the synthetics themselves, or instruct the intelligence to do the same? Thus they could let organics learn from their mistakes and make machines that didn't cause war.
2.) If the crucible can kill all synthetic life, why doesn’t it just do that instead of harvesting organics for a billion years? Again, this would let the organics live and learn from their mistakes.
3.) Why doesn’t the Reaper Intelligence and/or the Leviathans take notice of exceptions, like the organic/synthetic symbiotes Javik mentions or the peace between Geth and Quarians?
4.) Why did they hide if they can kill reapers?
Any insight is appreciated!!!
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