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.