Just cut off the brown bit and use the rest.

Of course.

Trim away the brown and carry one.

Yes ... it is quite safe to eat the 'brown edges' where you cut the cabbage.

It's just oxiditation.

You may like to invest in a plastic knife, for cutting cabbage or lettuce.

It does actually reduce the 'browning' quite a bit.

just cut the brown stuff off.  this happens when you cut a part off and save the rest for later.  the browning is oxidation.  some fruits and vegetables oxidize faster/worse than others 

Oxidation does not mean spoiled beyond use. 

sure you can cut off the oxidized portions and eat the rest.  There is nothing harmful about leaving the brown bits and cooking the rest of the cabbage either.  

the brown is just not aesthetically pleasing. 

Yes you can just a little oxidation at the cuts. Doesn't look as nice on the plate, but no harm.

yes. it will be fine

probably not. . .. . .