Problem 14
Let \(f: A \rightarrow B\) be a function, and \(Y \subseteq B\). Prove or disprove: \(f^{-1}\left(f\left(f^{-1}(Y)\right)\right)=f^{-1}(Y)\).
Problem 15
This question concerns functions \(f:\\{A, B, C, D, E, F, G\\} \rightarrow\\{1,2,3,4,5,6,7\\} .\) How many such functions are there? How many of these functions are injective? How many are surjective? How many are bijective?
Problem 16
This question concerns functions \(f:\\{A, B, C, D, E\\} \rightarrow\\{1,2,3,4,5,6,7\\} .\) How many such functions are there? How many of these functions are injective? How many are surjective? How many are bijective?
Problem 17
This question concerns functions \(f:\\{A, B, C, D, E, F, G\\} \rightarrow\\{1,2\\} .\) How many such functions are there? How many of these functions are injective? How many are surjective? How many are bijective?
Problem 18
Prove that the function \(f: \mathbb{N} \rightarrow \mathbb{Z}\) defined as \(f(n)=\frac{(-1)^{n}(2 n-1)+1}{4}\) is bijective.
