(Solved): Suppose S is a set and A is a subset of S. DEFINITION 1.1. The characteristic function of a subset ...

Suppose $S$ is a set and $A$ is a subset of $S$. DEFINITION 1.1. The characteristic function of a subset $A$ of $S$ is the function $f_A:S\to \{0,1\}$ so that $f_A(x)=\{\begin{array}{ll}1& \text{ if }x\in A\\ 0& \text{ if }x\notin A\end{array}$ (a) Consider the characteristic function $f_A:\{1,2,3\}\to \{0,1\}$ for the subset $A=\{3\}$ of the set $S=\{1,2,3\}$. Draw a Venn diagram for $f_A$, labelling the following sets: - $f(\{3\})$ - $f^{-1}(\{1\})$ - $f^{-1}(\{0\})$. (b) Let $S$ be a set and suppose $g:S\to \{0,1\}$ is a function: prove that it is the characteristic function of some subset of $S$, i.e. there exists some subset $A\subseteq S$ such that $g=f_A$.