Multiprogramming is a form of parallel processing on a single processor - or uniprocessor. Basically the operating system executes part of a one program, then part of another, and so on. It does it so quickly it seems to you and I that the programs are all being executed at once when in fact they are executed one at a time in parts to appear that way.
Timesharing is a very old system that basically used a special terminal that connected to a Central computer that offered a full operating system for the user to take advantage of. The users were charged for the time they were connected, the files stored and I believe a flat rate per month on top of that.
Multitasking is schedules what processes or tasks are to be running at any given time as well as when a waiting process will be scheduled to run. This allows the illusion of parrell processing, much like multiprogramming.
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