(2) Let f: X rightarrow Y be a function and let B subseteq Y. Prove that f^{-1}(Y-B)= X-f^{-1}(B).

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Step:1 ：Let x in f^{-1}(Y-B). This means that f(x) in Y-B. By the definition of set difference, this means that f(x) in Y and f(x) notin B. Since f(x) notin B, x notin f^{-1}(B). Therefore, x in X - f^{-1}(B).Step:2 ：Conversely, let x in X - f^{-1}(B). This means that x in X and x notin f^{-1}(B). Since x notin f^{-1}(B), f(x) notin B. Therefore, f(x) in Y - B, which means that x in f^{-1}(Y-B).Step:3 ：(boxed{f^{-1}(Y-B) = X - f^{-1}(B)})

Answer

Step: 1 ：Let x in f^{-1}(Y-B). This means that f(x) in Y-B. By the definition of set difference, this means that f(x) in Y and f(x) notin B. Since f(x) notin B, x notin f^{-1}(B). Therefore, x in X - f^{-1}(B).Step: 2 ：Conversely, let x in X - f^{-1}(B). This means that x in X and x notin f^{-1}(B). Since x notin f^{-1}(B), f(x) notin B. Therefore, f(x) in Y - B, which means that x in f^{-1}(Y-B).Step: 3 ：(boxed{f^{-1}(Y-B) = X - f^{-1}(B)})

(2) Let $f: X \rightarrow Y$ be a function and let $B \subseteq Y$. Prove that $f^{-1}(Y-B)=$ $X-f^{-1}(B)$.

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