### Arithmetic Operations on Functions from Sets into Functional Sets

Artur Korniłowicz
2009 Formalized Mathematics
In this paper we introduce sets containing number-valued functions. Different arithmetic operations on maps between any set and such functional sets are later defined. (p), 1898-9934(e) artur korniłowicz (Def. 4) If x ∈ X, then x is a real-valued function. Let us consider X. We say that X is rational-functions-membered if and only if: Let us consider X. We say that X is integer-functions-membered if and only if: (Def. 6) If x ∈ X, then x is an integer-valued function. Let us consider X. We say
more » ... consider X. We say that X is natural-functions-membered if and only if: (Def. 7) If x ∈ X, then x is a natural-valued function. One can check the following observations: * every set which is natural-functions-membered is also integer-functionsmembered, * every set which is integer-functions-membered is also rational-functionsmembered, * every set which is rational-functions-membered is also real-functionsmembered, * every set which is real-functions-membered is also complex-functionsmembered, and * every set which is real-functions-membered is also extended-realfunctions-membered. Let us mention that every set which is empty is also natural-functionsmembered. Let f be a complex-valued function. Observe that {f } is complex-functionsmembered. One can verify that every set which is complex-functions-membered is also functional and every set which is extended-real-functions-membered is also functional. One can verify that there exists a set which is natural-functions-membered and non empty. Let X be a complex-functions-membered set. One can verify that every subset of X is complex-functions-membered. Let X be an extended-real-functions-membered set. Note that every subset of X is extended-real-functions-membered. Let X be a real-functions-membered set. Note that every subset of X is real-functions-membered. Let X be a rational-functions-membered set. Observe that every subset of X is rational-functions-membered. Let X be an integer-functions-membered set. Note that every subset of X is integer-functions-membered. arithmetic operations on functions from sets . . . artur korniłowicz (6) N-Funcs X is a subset of N-PFuncs X. Let us consider X. One can verify the following observations: * C-PFuncs X is complex-functions-membered, * C-Funcs X is complex-functions-membered, * R-PFuncs X is extended-real-functions-membered, * R-Funcs X is extended-real-functions-membered, * R-PFuncs X is real-functions-membered, * R-Funcs X is real-functions-membered, * Q-PFuncs X is rational-functions-membered, * Q-Funcs X is rational-functions-membered, * Z-PFuncs X is integer-functions-membered, * Z-Funcs X is integer-functions-membered, * N-PFuncs X is natural-functions-membered, and * N-Funcs X is natural-functions-membered.