Question: Functions For these problems, let SN be the set of numbers {1,2, 3,…, N) 1) How many functions …
Show transcribed image text Functions For these problems, let SN be the set of numbers {1,2, 3,…, N) 1) How many functions are there that map SN to SN? 2) Argue that if a map f SN Sv is surjective, then f is a bijection. 3) Argue that if a map f : SN S is injective, then f is a bijection. 4) How many bijections are there that map SN to SN? 5) Suppose f is a map from a set S to itself, f S S. If S is a finite set, argue that |f(S) IS] if and only if f is a bijection. . Give an example of an infinite S and an f such that f is not a bijection, but If(S)|-|S| . f is a bijection, but f (S) is a proper subset of S
Functions For these problems, let SN be the set of numbers {1,2, 3,…, N) 1) How many functions are there that map SN to SN? 2) Argue that if a map f SN Sv is surjective, then f is a bijection. 3) Argue that if a map f : SN S is injective, then f is a bijection. 4) How many bijections are there that map SN to SN? 5) Suppose f is a map from a set S to itself, f S S. If S is a finite set, argue that |f(S) IS] if and only if f is a bijection. . Give an example of an infinite S and an f such that f is not a bijection, but If(S)|-|S| . f is a bijection, but f (S) is a proper subset of S
![Functions For these problems, let SN be the set of numbers {1,2, 3,..., N) 1) How many functions are there that map SN to SN? 2) Argue that if a map f SN Sv is surjective, then f is a bijection. 3) Argue that if a map f : SN S is injective, then f is a bijection. 4) How many bijections are there that map SN to SN? 5) Suppose f is a map from a set S to itself, f S S. If S is a finite set, argue that |f(S) IS] if and only if f is a bijection. . Give an example of an infinite S and an f such that f is not a bijection, but If(S)|-|S| . f is a bijection, but f (S) is a proper subset of S](https://media.cheggcdn.com/media%2F50d%2F50d9405b-354e-4f27-b5ef-554dc03984e1%2Fimage.png)