# Proof involving functions and intersection

1. Mar 28, 2012

### PCSL

Prove that $$f(\cap T_\alpha)=\cap f(T_\alpha)$$ for all choices of $$(T_\alpha) \alpha \in \lambda$$ if and only if f is one-to-one.

I've been working on this on and off for a day and have nothing to show for it... Any help pointing me in the right direction would be appreciated.

Also, more important then this question, can any of you recommend additional resources to review this type of stuff, mostly functions, order isomorphisms, etc. The book I am using does not give as many examples as I'd like.

2. Mar 28, 2012

### Dick

It's not really that hard. First assume f is one-to-one. Now can you show that if y is an element of $f(\cap T_\alpha)$ then it is an element of $\cap f(T_\alpha)$ and vice versa? For the opposite direction I'd do the contrapositive. Show if f is not one-to-one then you can find a counterexample.