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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?

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    $\begingroup$ Plug in the numbers $0,1,2,3,4,5$ for $n$ into $n+3$ to get the entries of the set. Do the same for $8-n$, and you see that you get the same entries, only in another order. So the sets coincide, although $n+3$ is never the same as $8-n$, if $n$ is a natural number. $\endgroup$ – Peter Aug 17 '17 at 11:26
  • $\begingroup$ I want to say something clever about the clear parallel to curly brackets and scopes in programming. I just don't know how to formulate it nicely. $\endgroup$ – Arthur Aug 17 '17 at 11:26
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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.

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