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threeder
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Homework Statement
Let [itex]f:X \rightarrow Y[/itex] and [itex]B_1, B_2 \in P(Y)[/itex] where P(Y) is the power set.
Prove that [itex]f^{-1}(B_1\cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)[/itex]
Homework Equations
The book gives this definition:
Suppose [itex]f:X \rightarrow Y[/itex] is a function.
The function [itex]f^{-1}:P(Y) \rightarrow P(X)[/itex] is defined by [tex]f^{-1}(B)=\begin{cases} x\in X | f(x)\in B \end{cases}\}[/tex] for [itex]B\in P(Y)[/itex]
The Attempt at a Solution
All I can do is just basically rewrite the definition:
Say [itex]y_0 \in B_1\cap B_2[/itex]. Then [tex]f^{-1}(\{y_0\})=\{x \in X ~|~ y_0=f(x)\}[/tex]
Then I make magic leap concluding that since [itex]y_0 \in B_1\cap B_2[/itex], [itex]y\in B_1[/itex] and [itex]y\in B_2[/itex]. Hence, these sets will be equal and so [itex]f^{-1}(B_1\cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)[/itex] . But I realize that a lot of grammar is missing. Can somebody help me out?