What does Terence want to say in Example 3.1.33? 
book
He says that $\left\{{n+3:n\in N, 0\leq n \leq 5  }\right\}$ is not the same as $n+3$ and I think that this is obvious. But I also think he tries to tell something else. Can you explain what he wants to say? 
 A: He shows why it is important to be exact in math.
It is $$\{n+3:n∈N,0\leqslant n\leqslant 5\} = \{8-n:n∈N,0\leqslant n\leqslant 5\},$$
but that does not imply 
$$n+3 = 8 - n,\qquad \text{ for } n∈ℕ, 0\leqslant n \leqslant 5.$$
The reason why this example works is that sets ignore the order of their elements.
So while $$\tilde{n}+3 \neq 8-\tilde{n},$$
for a specific $\tilde{n}∈ℕ$, $0\leqslant\tilde{n}\leqslant5$, 
the following is true:
For every $n∈ℕ, 0\leqslant n\leqslant 5$ there exists a $m∈ℕ, 0\leqslant m\leqslant 5$, such that 
$$n+3 = 8-m.$$

Edit: I really like this Example, because it illustrates the problem of abstraction that everyone faces in math - all the time. 
In the first grade you abstract from "1 apple + 2 apple = 3 apple" to "1+2=3". 
In the first term you abstract e.g. from "functions like $f(x)$" to general mappings. 
A very common question you hear in the first term, is: Lecturer writes  $$M=\{m∈ℝ | ...\} \\ \text{Let } n∈M \quad ...$$
"Why is $n∈M$, shouldn't it be $m∈M$?"
This is one moment of uncountable moments one faces during their studies. Some might only take fragments of a second to overcome, some might take years until it makes click and you suddenly understand. 
