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Simplify expressions using set-theoretic laws (properties of operations on sets):

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!

Simplify expressions using set-theoretic laws (properties of operations on sets): \(\overline{A \cap \overline{B}} : \overline{A \cap B} \cap V\).

Explore how to simplify set-theoretic expressions like \(\overline{A \cap \overline{B}} : \overline{A \cap B} \cap V\) using laws such as De Morgan's, set difference properties, and intersections. This solution provides a step-by-step breakdown suitable for undergraduate mathematics students.

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