Reference no: EM131724876

Show that the only bijections f: Z→ Z satisfying the following condition:

for all a, b ∈ Z : f (a+b) = f(a) + f(b) (*)

are f(x) = x and f(x) = -x.

(1) Show that the functions f(x) = x and f(x) = -x satisfy condition (*).

(2) By substituting the appropriate values for a and b, show that if f: Z→ Z satisfies (*), then f(0) = 0. Use it to deduce that f(-n) = -f(n) for every natural number n.

(3) Use induction to show that for every natural number n and every function f: Z→ Z satisfying (*), f(n) = n * f(1). Deduce that the same holds for negative integers.

(4) Use the fact that f is surjective to show that for every f: Z→ Z satisfying (*): f(1) = 1 or f(1) = -1, and conclude your solution.