 Let 𝐴 = {𝑥 ∶ 𝑥 ∈ ℝ, 𝑥 ≥ 10} and B = {𝑥: 𝑥 ∈ ℝ , 𝑥 ≥ 15}. Then 𝐴 ∩ 𝐵

To find the intersection of sets \( A \) and \( B \), we need to identify the elements that are common to both sets.

Given:
- \( A = \{x \mid x \in \mathbb{R}, x \geq 10\} \)
- \( B = \{x \mid x \in \mathbb{R}, x \geq 15\} \)

The intersection of \( A \) and \( B \) is denoted by \( A \cap B \), which represents the set of elements that are in both \( A \) and \( B \).

Since \( B \) includes all real numbers \( x \) that are greater than or equal to 15, and \( A \) includes all real numbers \( x \) that are greater than or equal to 10, the intersection will be the set of all \( x \) that are greater than or equal to 15, as this is the more restrictive condition.

Therefore:

\[
A \cap B = \{x \mid x \in \mathbb{R}, x \geq 15\}
\]

### Do you want details or have any questions?

Here are some related questions you might be interested in:
1. What is the union of sets \( A \) and \( B \)?
2. How would the intersection change if \( B = \{x \mid x \in \mathbb{R}, x \geq 20\} \)?
3. What is the complement of the intersection \( A \cap B \) in the set of real numbers?
4. How does the intersection change if \( A \) and \( B \) are defined over a different domain, like integers \( \mathbb{Z} \)?
5. What happens to \( A \cap B \) if \( A = \{x \mid x \in \mathbb{R}, x \geq 5\} \) and \( B \) remains the same?
6. What is the difference between \( A \) and \( B \) sets, \( A - B \)?
7. How do you express the intersection of three sets, \( A \cap B \cap C \)?
8. What is the symmetric difference between sets \( A \) and \( B \)?

**Tip:** When working with set operations, it is helpful to visualize the sets on a number line or using a Venn diagram to better understand their relationships.

Explore how to find the intersection of sets A and B in real numbers. Understand set theory concepts and learn through a detailed explanation. Suitable for advanced high school students and anyone interested in set theory operations.

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