Reply to Ableton Live and Traktor Pro 2

SO i have recently bought Traktor Pro 2 and looking to buy Ableton Live.

My question is i have bought Traktor on my PC and im looking on getting a MAC just after christmas. For my MAC that im buying would i have to buy Traktor all over again? Or is there a way i can download it to my MAC (i have downlaod version) And wat about Ableton if i buy the disc could i install it both on MAC and my PC?