Proof: Intersection, Inverse Function

• Mar 20th 2011, 09:08 PM
lfroehli
Proof: Intersection, Inverse Function
For this proof, we are to show that each containment is a subset of the other.

Part 1: $\displaystyle f^{-1}(\bigcap_{\lambda\in\Lambda}B_{\lambda}) = \bigcap_{\lambda\in\Lambda}f^{-1}(B_{\lambda})$.

Part 2: $\displaystyle f^{-1}(\bigcup_{\lambda\in\Lambda}B_{\lambda}) = \bigcup_{\lambda\in\Lambda}f^{-1}(B_{\lambda})$.

I honestly have no idea how to get started on this...any insight is appreciated!
• Mar 20th 2011, 11:14 PM
FernandoRevilla
It is almost routine knowing the definition of $\displaystyle f^{-1}(B)$ . For example:

$\displaystyle x\in f^{-1}(\bigcap_{\lambda\in\Lambda}B_{\lambda})\Rightar row f(x)\in \bigcap_{\lambda\in\Lambda}B_{\lambda}\Rightarrow f(x)\in B_{\lambda}\;\forall \lambda\in\Lambda\Rightarrow$

$\displaystyle x\in f^{-1}(B_{\lambda})\;\forall \lambda\in\Lambda \Rightarrow x\in \bigcap_{\lambda\in\Lambda}f^{-1}(B_{\lambda})$

etc.
• Mar 21st 2011, 12:48 AM
emakarov
This question was also discussed in this thread.