Let's work through the questions one by one.

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: $A \cap B^c = \{3, 5, 7\}$

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 $n(A \cup B) = 49$

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: $39 \leq x \leq 63$

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: $A = B = \emptyset$

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: $A = B$

Would you like further details on any of these answers?

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.