Up until fitted carpets came onto the scene, a carpet was a very large rectanglular flooring which filled most of the room (it was usually layed on top of lino.) A rug is a smaller version and is now placed on fitted carpets or on floorboards. A mat is something we can pick up easily for example in the bathroom, at the front door.
This reminds me of 'The Queen and I' when the royal family had to cut their carpets down to fit into their council house...!!!!
A friend worked for the BBC. He was given a new office. Midway through his first day, a gang of workmen turned up and carefully cut two feet off all sides of the wall-to -wall carpet. When he asked why, he was told that he was one grade below the rank that was entitled to fitted carpet. His grade was only entitled to a rug. When he questioned the wisdom of this, he got the reply "Oh, and you're not entitled to a standard lamp either", and the men took that away too.
A carpet is what you are hauled onto when you've been bad.
A rug is what's pulled from under you when your mate doesn't confirm your alibi.
(alternatively a rug is bright orange and worn on the head by baldies)
Mat is the paint on a blackboard.
Depends where you live I think. My Americans friends all have rugs and don't recognise the word carpet. I have always thought a mat was some thin, slightly scruffy thing that you wiped you feet on when you came in the door.
Fred, I remember when I worked in the brewery in a former life. The thickness of the carpet in your office depended on your grad - directors got the squishiest. I remember one bloke feeling under the door with a tape measure to try to determine what grade his boss was.....