I'm trying to see if there is any intuition pump / analogy that allows me to internalize ( and readily derive them) a series of results about the concepts of composition mixed with injectivity/surjectivity.
Propositions
Consider two functions $f:A\to B$ , $g:B \to C$.
I have the following propositions regarding injectivity,surjectivity and composition of functions.
i) $g \circ f$ injective $\to$ $f$ injective
ii) $g \circ f$ surjective $\to$ $g$ surjective
iii) $f$ and $g$ injective $\to$ $g \circ f$ injective
iv) $f$ and $g$ surjective $\to$ $g \circ f$ surjective
v) $g$ not injective and $f$ surjective $\to$ $g \circ f$ not injective.
Non injectiveness is preserved as long as the first function is surjective.
vi) $f$ not surjective and $g$ injective $\to$ $g \circ f$ not surjective.
Non surjectiveness is preserved as long as the second function is injective.
Internalization
I understand the concept of composition, injectivity and surjectivity perfectly ( both intuitively as well as formally ).
I also understand each of those propositions perfectly.
I can prove each one of them and i have an intuition for each one of them.
But i still don't have an intuition for the general mixing of composition with injectivity/surjectivity.
I feel that the only thing which can lead me to such results ( not their proofs, which i know how to obtain from the results ) is memorization and it is a bit hard to memorize them.
I'm thinking however,whether there is a simple way to understand the mixing of composition with injectivity that makes each of those results blatantly and imediately obvious.
I'm looking for a neat concept/analogy/intuition pump about the mixing of composition together with injectivity/surjectivity that encapsulates its essence.
The mentioned intuition pump would lead anyone possessing it to each one of those results ( at the moment, the only thing that can lead me to each one of those results is memorization ).
To give an example about the thing i'm looking for ...
Before learning about associated digraphs, learning about all those properties about relations ( reflexivity, symmetry, transitivity ) was a bit painful.
Before learning about the geometric interpretation of complex numbers, learning all their properties was a lot harder.
Before learning the concepts of "eventually" and "frequently" in the context of sequences ( analysis ) , it was a lot harder to understand the notions of convergence and limit.
Before learning the notion of "equinumerosity relation" in the context of cardinality of sets, it was a lot harder to make sense of it.
There are other examples as well.
In summary, i'm curious if anyone has encapsulated in his(her) mind each one of those results about composition/injectivity/surjectivity, not by means of memorization, but by an intuitive and obvious intuition pump/analogy that discards any need for memorization.