Let's analyze the given sets $A$ and $B$:

• Set $A$: $A = \{ |x| : x \in \mathbb{R} \}$
  This set represents all possible values of the modulus (absolute value) of real numbers. Since the modulus of any real number $x$ is always non-negative, the set $A$ includes all non-negative real numbers: $A = [0, \infty)$

• Set $B$: $B = \{ x : x \in \mathbb{R} \}$
  This set represents all real numbers. Therefore: $B = (-\infty, \infty)$

Comparing the Sets $A$ and $B$:

1. Subset Relation:

   • Clearly, every element of $A$ (which is non-negative) is also an element of $B$ (which includes all real numbers). Therefore, $A \subseteq B$.
   • However, not every element of $B$ is in $A$, since $B$ includes negative numbers, which are not in $A$. Hence, $B \not\subseteq A$.

2. Equality:

   • The sets $A$ and $B$ are not equal because $A$ only contains non-negative numbers, whereas $B$ contains both negative and positive numbers.

So, the relationship between $A$ and $B$ is: $A \subset B$ $A$ is a proper subset of $B$.

Would you like any further details or have any questions?

Here are five related questions you could explore:

1. What is the union of sets $A$ and $B$ in this context?
2. How would the set $A$ change if defined as $A = \{ |x| : x \in \mathbb{Z} \}$, where $\mathbb{Z}$ is the set of integers?
3. What is the intersection of sets $A$ and $B$?
4. Can you define a set $C$ such that $B \subset C$?
5. How would the set $A$ look if it were defined as $A = \{ |x| : x \in [0, 1] \}$?

Tip: When working with sets, especially when defining them in terms of properties like modulus, always consider the range of possible values to understand the subset and equality relationships.