Default Rogers with AT&T/T-Mobile

Hi,
I have done some searching but couldn't come across with a firm answer, So can you guys help me out?

I'm currently with Rogers, if I purchased an unlocked phone from someone or ebay.
Will the frequencies work?

So can I put a Rogers sim card into an unlocked AT&T or T mobile phone and have it fully functioning?

Before posting this thread, my assumption was that AT&T is fully compatible with Rogers (voice, 3G) , but on the other hand Rogers 3G doesn't work on T mobile phones?

Your answer is very appreciated!