please give complete justification for each of the assertion

please give complete justification for each of the assertions in the proof.

Let N represent the set of positive integers. Find all functions f: N vector N such that f(1) = 2 and f(xy) = f(x)f(y) - f(x + y) + 1 for all x, y element of N and using Mathematical Induction, prove your answer is correct.

Solution

ans

we know that, f(xy)=f(x)f(y)- f(x+y)+1

now f(x+y)=1-f(xy)+f(x)f(y)

now take x=k, y=1

f(k+1)=1-f(k)+f(1)f(k)

also f(1)=2

so,

f(k+1)=1+f(k).