Question: Functions For these problems, let Sy be the set of numbers [1, 2,3,N. 1) How many functions are t…
Show transcribed image text Functions For these problems, let Sy be the set of numbers [1, 2,3,N. 1) How many functions are there that map Sy to Sn? 2) Argue that if a map f:Sv Sv is surjective, then f is a bijection. 3) Argue that if a map f:SvSy is injective, then f is a bijection. How many bijections are there that map Sy to Sy? 5) Suppose f is a map from a set S to itself, f : s → s. * If S is a finite set, argue that If(s)|-Isl 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)-Sl . f is a bijection, but f(S) is a proper subset of S
Functions For these problems, let Sy be the set of numbers [1, 2,3,N. 1) How many functions are there that map Sy to Sn? 2) Argue that if a map f:Sv Sv is surjective, then f is a bijection. 3) Argue that if a map f:SvSy is injective, then f is a bijection. How many bijections are there that map Sy to Sy? 5) Suppose f is a map from a set S to itself, f : s → s. * If S is a finite set, argue that If(s)|-Isl 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)-Sl . f is a bijection, but f(S) is a proper subset of S
