Hello, I would like to ask if somebody has any site or materials that would explain these kinds of exercises.
Examples:
Let $A = \{1,2,3,4,5,6,7\}$ and $B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$. How many injective mappings $f$ from set $A$ to set $B$ are there, such that $f^{-1}$ ($\{1, 2, 3, 4, 5, 6\}) = \{1,2,3,4\}$?
Let $A = \{1,2,3,4,5,6,7,8,9,10,11\}$ and $B = \{1,2,3,4,5,6,7\}$. How many mappings $f$ are there from set $A$ to set $B$ such that $f(\{1,2,3,4,5,6\}) \supseteq \{1,2,3,4\}$ ?
I know how to solve these when they are injective but I dont know how to do it for example surjective or if its just a function (like in second example).
And also what does change in counting them if they are $=$, $\subseteq$ or $\supseteq$.
I would be thankful for any help/explanation or sites/materials that explain how to solve this.
The comments to your question are excellent.
Also, Richard Stanley's twelvefold way [that's where I have first seen it] counts functions between two sets under the constraints of labeled and unlabeled sets and all functions, injective functions, surjective functions etc. This gives a good general framework to such questions. Binomial coefficients, Stirling numbers and falling factorials make an appearence.
See https://dlmf.nist.gov/26.17
I prefer this representation to that at Wikipedia where group theoretic language is used regarding labeling. It's in Stanley's Enumerative Combinatorics book (see https://math.mit.edu/~rstan/ec/) as well as probably in his lecture notes on his website somewhere.