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Exercise 14 - Chapter 3

edited June 2018

In Example 3.12 we identified the paths of the loop graph (3.13) with numbers $$n \in \mathbb{N}$$ . Paths can be concatenated. Given numbers $$m, n \in \mathbb{N}$$ , what number corresponds to the concatenation of their associated paths?

$$m+n$$. Since $$n=s^n(0)$$ and $$m=s^m(0)$$, concatenation gives $$s^{n}(s^m(0))=s^{(n+m)}(0)=n+m$$. So, concatenation corresponds to summation, and is commutative (since there is only one symbol/function being concatenated).
Comment Source:\$$m+n\$$. Since \$$n=s^n(0)\$$ and \$$m=s^m(0)\$$, concatenation gives \$$s^{n}(s^m(0))=s^{(n+m)}(0)=n+m\$$. So, concatenation corresponds to summation, and is commutative (since there is only one symbol/function being concatenated).