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Let $\mathbb{Q}$ denote the set of rational numbers. Find all functions from $\mathbb{Q}$ to $\mathbb{Q}$ which satisfy

  1. $f(1)=2$; and
  2. $f(xy)=f(x)f(y)-f(x+y)+1$.

I know that $f(x)=x+1$ satisfies the equation but I don't know how to prove if this is the only such function. This question was in the 'Induction' section of my textbook.

Any hints?

Naweed G. Seldon
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  • Can you prove $f(0)=1$ ? that would be a start. You are given $f(1)$. Now try to find $f(2)$. – GEdgar May 14 '18 at 20:09
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    As a hint: the values of $f(n)$ and $f(1)$ determine the value of $f(n+1)$, by letting $x=n$ and $y=1$ in the functional equation. This allows you to prove $f(n)=n+1$ for all $n\ge 1$, by induction. – Mike Earnest May 14 '18 at 20:11

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