Function and subsets - inverses

[threeder]

Homework Statement

Let f:X \rightarrow Y and B_1, B_2 \in P(Y) where P(Y) is the power set.

Prove that f^{-1}(B_1\cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2)

Homework Equations

The book gives this definition:
Suppose f:X \rightarrow Y is a function.

The function f^{-1}:P(Y) \rightarrow P(X) is defined by f^{-1}(B)=\begin{cases} x\in X | f(x)\in B \end{cases}\} for B\in P(Y)

The Attempt at a Solution

All I can do is just basically rewrite the definition:

Say y_0 \in B_1\cap B_2. Then f^{-1}(\{y_0\})=\{x \in X ~|~ y_0=f(x)\}

Then I make magic leap concluding that since y_0 \in B_1\cap B_2, y\in B_1 and y\in B_2. Hence, these sets will be equal and so f^{-1}(B_1\cap B_2) = f^{-1}(B_1) \cap f^{-1}(B_2) . But I realize that a lot of grammar is missing. Can somebody help me out?

 

[HallsofIvy]

The "magic leap" can be given in more detail. You are basically trying to prove "A= B". To prove that for sets, first prove "A\subset B and the prove B\subset A. And to prove A\subset B, start with "if x\in A" and use the properties of A and B to conclude "then x\in B".

Here, you want to prove that f^{-1}(B_1\cap B_2)= f^{-1}(B_1)\cap f^{-1}(B_2). So start "if x\in f^{-1}(B_1\cap B_2)". Then, by definition of f^{-1}, there exist y\in B_1\cap B_2 such that f(x)= y. Since y is in B_1\cap B_2, y is in B_1 and y is in B_2.
1) If y is in B_1 and f(x)= y, then x\in f^{-1}(B_1).
2) If y is in B_2 and f(x)= y, then x\in f^{-1}(B_2).

Since both are true, x\in f^{-1}(B_1)\cap f^{-1}(B_2).
That is, f^{-1}(B_1\cap B_2)\subset f^{-1}(B_1)\cap f^{-1}(B_2).

Now, to prove the other way, that f^{-1}(B_1)\cap f^{-1}(B_2)\subset f^{-1}(B_1\cap B_2)/itex], start:<br /> If x\in f^{-1}(B_1)\cap f^{-1}(B_2) and I will let you try the rest.

 

[threeder]

Lets try it:

Say x\in f^{-1}(B_1)\cap f^{-1}(B_2). Then there exists y\in B_1 and it also y\in B_2 such that y=f(x). Because of x belonging to both sets, and such y existence, it also belongs to y\in B_1\cap B_2 such that y=f(x). Hence f^{-1}(B_1)\cap f^{-1}(B_2) is a subset of f^{-1}(B_1\cap B_2)

right?