How can we prove that the sum of two arbitrary functions is a function further more if ab belong to Z how can we prove that aF1 plus bF2 is a function?

Answer 1

I'm not sure I really understand the question. If you have two functions, F1 and F2, then by definition of function you have: If X=Y then F(x)=F(y) for both F1 and F2 (F1+F2)(x) is defined as F1(X) + F2(X), and for every X there exists a unique F1(x) and F(2)(x) and the sum of these two numbers is obviously unique. If the question meant that the range of F1 and F2 were included in Z and how do you show that the range of F1 + F2 is also a subset of Z, then it's as simple as: the sum of two integers is an integer. and for the second part: the product of two integers is an integer.