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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 $f_{A}:S→{0,1}$ so that $f_{A}(x)={10 ifx∈Aifx∈/A $ (a) Consider the characteristic function $f_{A}:{1,2,3}→{0,1}$ 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 $g:S→{0,1}$ is a function: prove that it is the characteristic function of some subset of $S$, i.e. there exists some subset $A⊆S$ such that $g=f_{A}$.

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