Question 175
Find all surjective functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all positive integers $a$ and $b$, exactly one of the following equations is true:
$$
\begin{aligned}
f(a) &= f(b) \\
f(a+b) &= \min \{f(a), f(b)\}
\end{aligned}
$$
Remarks: $\mathbb{N}$ denotes the set of all positive integers. A function $f: X \rightarrow Y$ is said to be surjective if for every $y \in Y$ there exists $x \in X$ such that $f(x)=y$.
$$
\begin{aligned}
f(a) &= f(b) \\
f(a+b) &= \min \{f(a), f(b)\}
\end{aligned}
$$
Remarks: $\mathbb{N}$ denotes the set of all positive integers. A function $f: X \rightarrow Y$ is said to be surjective if for every $y \in Y$ there exists $x \in X$ such that $f(x)=y$.