Linear order on the set N^N

Question

Give the example of the Linear order on the set: $\mathbb N^\mathbb N$

I know that $\mathbb N^\mathbb N$ is a function from $\mathbb N$ to $\mathbb N$ and it's a string, but I'm unable to give an example.

Answer 1

First, think of words in a dictionary: how do you order them? Which comes first: Apple or Orange? You compare the first letters and since A is before O, you say that Apple is before Orange. If the first letters, are the same then compare the second so apple is before avocado. If they are the same until one ends then you need a rule, e.g. the shorter one is first so grape is before grapefruit.

Your functions can be written as a list of numbers. $f(1), f(2), f(3), ... $ To compare $f$ and $g$, compare $f(1)$ and $g(1)$. If $f(1)$ is less then $f$ is less. If they are the same then compare $f(2)$ and $g(2)$, etc. Continue until you find a difference.