As the title says, I struggle to understand why ( Injections,∘ ) is not a group while ( Injections_finite,∘ ) is.
As I understand, there are 4 properties needed for set to be a group.
- Closed operation
For both statements, this is valid. Since these are functions, they are defined on the entire domain.
The identity function which takes every element to itself is the identity : f(x) -> x
It is verifiable as a result of closure.
This is what I have issues understanding.
- For ( Injections,∘ ), the explanation given is: Since the domain can have infinite size, it is not necessary that these functions are invertible.
- What does the infinite size of the domain have to do with the absence of inverse for functions that are part of that domain?
- For ( Injections_infinite,∘ ), the explanation given is: The existence of inverses is the result of the proposition which states that if the domain and co-domain are finite and have the same size, every injective function from one to the other is also a bijection. Since these are bijections, their invertability follows.
- Does "injective function from one to the other is also a bijection" mean that functions that are mapped from a finite domain to a finite co-domain are both injective and surjective?
- Why does their bijection imply invertability?
Thanks for the help in advance!!