Cutting and pasting polygons in QGIS so they do not overlap?

Question

I use QGIS, and I need to cut and paste polygons so they don't overlap. Here's what I mean: I have shapefiles for cities that I'm pasting into a new polygon shapefile. I also have shapefiles for counties, and I want to cut and paste the parts of the counties that are NOT part of a city.

Sometimes it will automatically paste it the way I want, as if it's reading my mind. Sometimes it will not. Other times, it will paste a much-simplified version of the polygon that's unrecognizable. There's no way to predict what it will do.

How do I make it so it ALWAYS pastes it as a non-overlapping polygon?

Answer 1

To get a shapefile with polygons of counties with gaps of your cities polygons use Symetrical difference in Vector -> Geoprocessing Tools.

If you want to create new shapefile from both shapefiles (not overlapping) use Union function Vector -> Geoprocessing Tools.