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Let's work through the questions one by one.

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### Q10. Set Theory
Given:
- \( U = \{1, 2, 3, \dots, 10\} \)
- \( A = \{x : x \text{ is prime}\} \)
- \( B = \{x : x \text{ is even integer}\} \)

We need to find \( A \cap B^c \).

**Solution:**
1. List elements:
   - **Primes in \( U \)**: \( A = \{2, 3, 5, 7\} \)
   - **Even integers in \( U \)**: \( B = \{2, 4, 6, 8, 10\} \)

2. Find \( B^c \) (elements in \( U \) but not in \( B \)):
   - \( B^c = \{1, 3, 5, 7, 9\} \)

3. Find \( A \cap B^c \):
   - Intersection \( A \cap B^c = \{3, 5, 7\} \)

**Answer:** \( A \cap B^c = \{3, 5, 7\} \)

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### Q11. Set Operations with Given Information
Let \( A \) and \( B \) be sets such that:
- \( n(A - B) = 14 + x \)
- \( n(B - A) = 3x \)
- \( n(A \cap B) = x \)

Tasks:
1. **Draw a Venn diagram** to illustrate this information.
2. **If \( n(A) = n(B) \)**, find:
   - (1) The value of \( x \)
   - (2) \( n(A \cup B) \)

**Solution:**
1. For \( n(A) = n(B) \), we know:
   \[
   n(A) = n((A - B) \cup (A \cap B)) = (14 + x) + x = 14 + 2x
   \]
   Similarly, \( n(B) = n((B - A) \cup (A \cap B)) = 3x + x = 4x \).
   
   Since \( n(A) = n(B) \), we have:
   \[
   14 + 2x = 4x
   \]
   Solving for \( x \):
   \[
   14 = 2x \implies x = 7
   \]

2. With \( x = 7 \):
   - \( n(A) = 14 + 2 \cdot 7 = 28 \)
   - \( n(B) = 4 \cdot 7 = 28 \)
   - \( n(A \cup B) = n(A) + n(B) - n(A \cap B) = 28 + 28 - 7 = 49 \)

**Answer:** \( x = 7 \) and \( n(A \cup B) = 49 \)

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### Q12. Probability of Watching News Channels
Given:
- 63% watch Channel A
- 76% watch Channel B
- \( x \)% watch both

We need to show \( 39 \leq x \leq 63 \).

**Solution:**
Using the formula for union of probabilities:
\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]
Here, \( P(A \cup B) \leq 100\%\), so:
\[
63 + 76 - x \leq 100
\]
\[
139 - x \leq 100 \implies x \geq 39
\]

Also, since \( x \leq P(A) \), we get:
\[
x \leq 63
\]

**Answer:** \( 39 \leq x \leq 63 \)

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### Q13. Probability with Equal Sets
Given \( P(A) = P(B) \) and \( P(A \cap B) = 0 \), show that \( A = B \).

**Solution:**
1. Since \( P(A) = P(B) \) and \( P(A \cap B) = 0 \), this implies \( A \) and \( B \) are mutually exclusive (cannot both occur).

2. However, if both sets have the same probability and cannot occur together, this generally means \( A = B = \emptyset \) (the empty set).

**Answer:** \( A = B = \emptyset \)

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### Q14. Set Equality
Let \( A \) and \( B \) be sets. If:
- \( A \cap B = B \cap A = \emptyset \)
- \( A \cup X = B \cup X \) for some set \( X \)

Show that \( A = B \).

**Solution:**
If \( A \cup X = B \cup X \) and both are disjoint with \( X \), it suggests that the only way to have the union equal is if \( A \) and \( B \) are identical in their contents.

**Answer:** \( A = B \)

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Would you like further details on any of these answers? 

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**Related Questions:**
1. How to determine complement and intersection of sets?
2. How to use the inclusion-exclusion principle in probability?
3. How can we represent set differences in Venn diagrams?
4. How to interpret probabilities for mutually exclusive events?
5. How does disjoint union affect set equality?

**Tip:** In Venn diagrams, use shading to clearly represent set differences, intersections, and complements for easier visualization.

A set of problems on set theory, probability, and equality. Topics include finding intersections with complements, determining elements of unions based on given conditions, analyzing probabilities for events, and proving set equality with unions.

Explore solutions to set theory problems involving union, intersection, and complements of sets, as well as probability with overlapping events. Topics include calculating set differences, analyzing probabilities for news channel viewership, and proving set equality. Ideal for high school students in Grades 9-12.

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