Describe all the subset relationships that hold for the sets given in Example.ExampleLet P...

Describe all the subset relationships that hold for the sets given in Example.

Example

Let P be a computer program that accepts an integer as input and produces an integer as output. Let A = B = Z. Then P determines a relation fP defined as follows: (m, n)  fP means that n is the output produced by program P when the input is m.

It is clear that fP is a function, since any particular input corresponds to a unique output. (We assume that computer results are reproducible; that is, they are the same each time the program is run.)

Example can be generalized to a program with any set A of possible inputs and set B of corresponding outputs. In general, therefore, we may think of functions as input-output relations.