Another crazy question.
Is this a countable set; $$A = \{-4, -2, -1, 0, 1, 2, 3, 4\}$$
I think it is. But my teacher says it isn’t due to the concept , he says, is of one to one mapping with a some subset of natural number. Since some elements there contain non - $\Bbb N$ elements.
From what I know, a countable set is either a finite set, or a set with the same cardinality with some subset of $\mathbb N$, which make me think it is.
On
You are correct and what your teacher says (at least as you have represented it) is wrong. A countable set can have elements that are not in $\mathbb{N}$; you just need the set to have a bijection to a subset of $\mathbb{N}$ (which does not need to be the identity map!). In other words, whether a set is countable depends only on how many elements it has, not what those elements are.
A countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.
Just verify that $f: A \to \{1, \ldots, 8 \}$
$$f(0)=1, f(1)=2, f(2)=3, f(3)=4, f(4)=5, f(-4)=6, f(-2)=7, f(-1)=8 $$
is a bijection and you are done.
p/s: Of course, all finite set are countable.