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(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 fA:S→{0,1} so that fA(x)={10 if x∈A if x∈/A (a) Consider the characteristic function fA:{1,2,3}→{0,1} for the subset A={3} of the set S={1,2,3}. Draw a Venn diagram for fA, 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=fA.