Let f and g be functions from the integers to the integers P

Let f and g be functions from the integers to the integers. Prove that if both functions are surjections, then so their composition, but that their sum is not necessarily a surjection.

Solution

f: Z to Z and

g: z to z

Both f and g are surjections.

To prove that f+g is not a surjections

Let us give example as

f(x) = x for all x in Z

and g(x) = -x for all x in Z

(f+g)(x) = x-x =0 for all x in Z

Hence only 0 is the element in the range so f+g cannot be onto though

f and g are onto

Also fog(x) = -x = gof(x) is onto

But f+g or g+f is not onto