The given expression is:

$\frac{\overline{A \cap \overline{B}}}{\overline{A \cap B}} \cap V$

We can simplify it step by step using standard set-theoretic laws, such as De Morgan's Laws, properties of complements, intersections, and set differences. Let's proceed with the simplification:

Step 1: Understanding the expression

We can interpret the fraction $\frac{\overline{A \cap \overline{B}}}{\overline{A \cap B}}$ as a set difference. In set theory, this is equivalent to:

$\overline{A \cap \overline{B}} \setminus \overline{A \cap B}$

Now, we have:

$(\overline{A \cap \overline{B}} \setminus \overline{A \cap B}) \cap V$

Step 2: Simplify using De Morgan’s Law

De Morgan's Law states:

$\overline{A \cap B} = \overline{A} \cup \overline{B}$

Applying this to both terms:

1. $\overline{A \cap \overline{B}} = \overline{A} \cup B$
2. $\overline{A \cap B} = \overline{A} \cup \overline{B}$

Now the set difference becomes:

$(\overline{A} \cup B) \setminus (\overline{A} \cup \overline{B})$

Step 3: Simplify the Set Difference

The set difference $X \setminus Y$ is equivalent to the intersection of $X$ with the complement of $Y$:

$(\overline{A} \cup B) \cap \overline{(\overline{A} \cup \overline{B})}$

Applying De Morgan's Law again to the complement of $\overline{A} \cup \overline{B}$:

$\overline{(\overline{A} \cup \overline{B})} = A \cap B$

Thus, the set difference simplifies to:

$(\overline{A} \cup B) \cap (A \cap B)$

Step 4: Apply Distribution of Intersection

Now distribute the intersection over the union:

$(\overline{A} \cap (A \cap B)) \cup (B \cap (A \cap B))$

Simplifying each part:

1. $\overline{A} \cap (A \cap B) = \varnothing$ (since $A$ and $\overline{A}$ are disjoint)
2. $B \cap (A \cap B) = A \cap B$

Thus, the expression reduces to:

$A \cap B$

Step 5: Intersection with $V$

Now we include the final intersection with $V$:

$(A \cap B) \cap V$

This is simply:

$A \cap B \cap V$

Final Simplified Expression:

The fully simplified expression is:

$A \cap B \cap V$

Would you like further details or have any questions on this simplification?

Here are 5 related questions that build on this:

1. How does De Morgan’s law work for union operations?
2. What happens when we intersect a set with its complement?
3. How would the result change if $V = A$?
4. What is the difference between a set union and a set intersection?
5. How can we express set difference in terms of union and intersection?

Tip: Set difference can always be rewritten as an intersection with the complement of the second set!