How does multithreading take place on a computer with a single CPU?

The operating system's task scheduler allocates execution time to multiple tasks. By quickly switching between executing tasks, it creates the impression that tasks execute sequentially.

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On a single processor, Multithreading generally occurs by time-division multiplexing ("time slicing") in very much the same way as the parallel execution of multiple tasks (computer multitasking): the processor switches between different threads. This context switching can happen so fast as to give the illusion of simultaneity to an end user

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