Is it right that $\{2x : x \in\mathbb{Z}, |x| < 4\} = \{−6,−4,−2,0,2,4,6\}$?

Question

I just started reading some book where it explicitly says that
$\left\{2x : x \in\mathbb{Z}, |x| < 4\right\} =$ $\left\{−6,−4,−2,0,2,4,6\right\}$

But that's how I tried to solve it and I had different answer:

$2x: x \in\mathbb{Z}$ which means, the set of all things in the form $2x$ such that $x$ is an element of $\mathbb{Z}$
$\left\{..., -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, ...\right\}$

Then we have $|x| < 4$, which scales down the previous set to a range of only $[-2, 2]$, so if the set contains $\left\{-6, -4, 4, 6\right\}$ they're all proved to be wrong because their absolutes are all $≥ 4$.

So the correct set should actually only be $\left\{-2, 0, 2\right\}$ according to my understanding.

But Page 16 of Book Of Proof by Richard Hammack has a different solution with no explanation

I'm quite new to these set expressions so please tell me if I interpreted anything wrong.

Answer 1

$x$ is said to be an integer number with absolute value less then $4$. Thus $x$ is an integer ranging from $−3$ to $3$. Thus your set (made of elements of the form $2x$) is indeed that of the book.

Answer 2

You are only told that $|x| <4$, not that $|2x| <4$. So $|2x|$ can go up to $8$ and the given answer is correct.

Answer 3

You did solve another problem: $$\{x \mid x \in\mathbb{Z}, |2x| < 4\}.$$ Here the solution is your solution $\{−2,0,2\}$.

But you should split the problem into two parts. First find the set of values for $x$: $$x \in S := \{x \mid x \in \mathbb Z, |x| < 4\}.$$ This are all integers with absolute value less than four: $x \in \{-3, -2, -1, 0, 1, 2, 3\}$.

Then map this set with the function $f(x) = 2x$: $$\{2x \mid x \in S\} = \{-6, -4, -2, 0, 2, 4, 6\}.$$