Why is $\mathbb N+\mathbb N+\mathbb N$ similar to this set?

Question

From the lecture I learned $\mathbb N+\mathbb N+\mathbb N$ is similar to $\{0,1/2,2/3,3/4,...,1,1+1/2,1+2/3,1+3/4,...,2,2+1/2,2+2/3,2+3/4,...\}$(The original question is to ask to find a subset of $\mathbb Q$ similar to $\mathbb N+\mathbb N+\mathbb N$). I understand this intuitively because And similarly for $\mathbb N+\mathbb N+\mathbb N$. But rigorously what is $\mathbb N+\mathbb N+\mathbb N$? From the usual definition isn't $\mathbb N+\mathbb N+\mathbb N$ already a subset of $\mathbb Q$? And I cannot think of a bijective function that maps $\mathbb N+\mathbb N+\mathbb N$ to $\{0,1/2,2/3,3/4,...,1,1+1/2,1+2/3,1+3/4,...,2,2+1/2,2+2/3,2+3/4,...\}$ Could you give me an example of this? Thank you.