I need to proof knowing that some two sets and have the same cardinality that sets and , which are sets of maps from some set to sets and respectively, also have the same cardinality.
So here's my attempt:
So we know that those two sets are equal, so that means that there's a bijection between them, like this:
I have to create a bijection between and to prove that they have the same cardinality. So let be this bijection.
is supposed to take a function that goes from to and return a function that goes from to .
for some equals meaning .
Let us construct a function that does this . This function takes from and returns an element in by assigning one to the result of by the bijection between sets and .
So in the end, a bijection between sets and is a function .
Is this even partly correct?