Sure, here's a mathematical problem involving the empty set, and I'll provide a detailed solution for it.

Problem:

Consider the sets $A = \{x \in \mathbb{Z} : x^2 - 4x + 3 = 0 \}$ and $B = \{x \in \mathbb{Z} : x^2 - 1 = 0 \}$. Find $A \cap B$ (the intersection of sets $A$ and $B$) and determine if it is an empty set or not.

Solution:

1. Find the elements of set $A$:

   The set $A$ is defined by the quadratic equation $x^2 - 4x + 3 = 0$. To find the elements of $A$, solve this equation:

   $x^2 - 4x + 3 = 0$

   We can factorize the quadratic equation:

   $(x - 1)(x - 3) = 0$

   This gives the solutions:

   $x = 1 \quad \text{and} \quad x = 3$

   Since $A$ consists of integers ($x \in \mathbb{Z}$), the set $A$ is:

   $A = \{1, 3\}$

2. Find the elements of set $B$:

   The set $B$ is defined by the quadratic equation $x^2 - 1 = 0$. To find the elements of $B$, solve this equation:

   $x^2 - 1 = 0$

   We can factorize it as:

   $(x - 1)(x + 1) = 0$

   This gives the solutions:

   $x = 1 \quad \text{and} \quad x = -1$

   Since $B$ also consists of integers ($x \in \mathbb{Z}$), the set $B$ is:

   $B = \{1, -1\}$

3. Find the intersection $A \cap B$:

   The intersection $A \cap B$ consists of all elements that are in both sets $A$ and $B$.

   $A = \{1, 3\}, \quad B = \{1, -1\}$

   The only common element between $A$ and $B$ is $1$.

   Thus,

   $A \cap B = \{1\}$

4. Determine if $A \cap B$ is an empty set:

   The set $A \cap B = \{1\}$ is not empty because it contains the element $1$.

Conclusion:

The intersection $A \cap B = \{1\}$ is not the empty set.

Would you like more details on this problem or have any questions?

Here are some related questions you might find interesting:

1. How do you determine the union of two sets?
2. What are the properties of set operations like union, intersection, and difference?
3. How does the concept of the empty set apply in solving systems of equations?
4. Can an intersection of infinite sets be an empty set?
5. What are the differences between sets and multisets?

Tip: When solving set problems, always consider the definitions and properties of the sets involved to correctly perform operations like intersection and union.