I want to connect my projector to my Mac Book pro. Projector has resolution 1920x1080 Full HD. My question is how to output the same resolution from MacBook. My Mac Book pro display has resolution 1280x800, I am not sure if I can make output 1920x1080.
According to everymac.com, your particular MacBook Pro will display 2560x1600 maximum resolution (Thunderbolt) and 1920x1200 via HDMI.
If your projector supports 1920x1080, and it being "Full HD" as you described, it will most likey have an HDMI port. All you will need is a quality mini Display Port (mdP) to HDMI adapter