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Set Theoryhard
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Let f:X→Yf: X \to Yf:X→Y be a function, and let AAA and BBB be subsets of YYY. Consider the following statements regarding the preimage f−1f^{-1}f−1: I. f−1(A∖B)=f−1(A)∖f−1(B)f^{-1}(A \setminus B) = f^{-1}(A) \setminus f^{-1}(B)f−1(A∖B)=f−1(A)∖f−1(B) II. f(f−1(A))=Af(f^{-1}(A)) = Af(f−1(A))=A III. f−1(A∩B)=f−1(A)∩f−1(B)f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B)f−1(A∩B)=f−1(A)∩f−1(B) Which of these statements must hold true for any function fff and any subsets A,B⊆YA, B \subseteq YA,B⊆Y?