Let $f:X\longrightarrow Y$ be a function, $A, A_1, A_2$ subsets of $X$ and $B, B_1, B_2$ subsets of $Y$.

Asked Feb 26 '18 at 05:25

Active Feb 26 '18 at 10:05

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a) Prove that $f^{-1}(Y)=X$

b) Prove that $f^{-1}(Y-B)=X-(f^{-1}(B))$

I could kind of see where I want to go with the first one though I'm not really sure where to start. I know that every element in $x$ gets mapped to $y$. So the inverse image of $Y$ must be $X$. I think that's right...

However the second proof I need help with entirely. Please and thanks.

edited Feb 26 '18 at 05:26

Hanul Jeon

27,376

asked Feb 26 '18 at 05:25

Luis Robles

Your attempt for the first is right. To prove the second you can try to show $x\in f^{-1}[Y-B] \iff x \in X-f^{-1}[B]$ for all $x\in X$. – Hanul Jeon Feb 26 '18 at 05:32
Possible duplicate of Overview of basic results about images and preimages – Arnaud D. Feb 26 '18 at 10:28

Let $f:X\longrightarrow Y$ be a function, $A, A_1, A_2$ subsets of $X$ and $B, B_1, B_2$ subsets of $Y$.

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