Let A={modulus x:x belongs to R}, B{x:x belongs to R} then

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.

Explore the relationship between sets A and B in set theory. Analyze the modulus of real numbers and understand how sets A and B differ. Suitable for advanced learners interested in set theory and real numbers.

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