Having trouble understanding notation: $A \subseteq \mathbb{R}$ bounded $\Leftrightarrow$ $|A|$ bounded where $|A| := \{|x||x\in\mathbb{A}\}$

Question

The first part is crystal: A, which is the subset of the real numbers set is bounded iff... but that's where I get lost.

As far as I'm aware, if $A$ is a set, then $|A|$ is the amount of elements it has. What exactly does the definition part mean? A redefinition of $|A|$?

Answer 1

Here, $|A|$ no longer stands for the cardinality of $A$. Instead, $A$ is the set obtained by taking the absolute value of all the elements in $A$. So, as you've written, $$ |A| := \left\{ |x| : x \in A \right\} . $$ Hence, $y \in |A|$ if and only if $y = |x|$ for some $x \in A$. By definition, $A$ is bounded if there exists $M \geq 0$ such that $$ |x| \leq M $$ for all $x \in A$. Therefore, when deciding if $A$ is bounded, we only care about the absolute value of elements in $A$. From here, it's immediate that $A$ is bounded if and only if $|A|$ is bounded.

Answer 2

The claim depends on a new, different meaning of the notation $|A|$, namely the one it defines after "where".

Is it perhaps clearer for you if we use a different notation?

Let $f : \mathcal P(\mathbb R)\to\mathcal P(\mathbb R)$ be the function defined by $$ f(A) = \{\,|x|\,\mid x\in A\}$$ Then for every $A\subseteq \mathbb R$, it holds that $A$ is bounded iff $f(A)$ is bounded.