Let’s work through the solutions systematically.

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10. Write in set-theoretic notations:

(i) $3 \in A$
(ii) $4 \notin B$
(iii) $C \subseteq D$
(iv) $P \cap Q = \emptyset$

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11. True or false, with reasons:

(i) $\{a\} = \{a, b, c\}$ — False. A set with only $a$ is not equal to a set containing $a, b, c$.
(ii) $a \in \{a, b, c\}$ — True. $a$ is one of the elements in the set.
(iii) $a \subset \{a, b, c\}$ — False. $a$ as an individual element is not a subset but an element of the set.
(iv) $a \notin \{a, b, c\}$ — False. As shown in (ii), $a$ is in the set.

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12. Operations on sets $A, B, C$:

$A = \{1, 2, 3, 4\}, B = \{2, 4, 5, 8\}, C = \{3, 4, 5, 6, 7\}$

(i) $A \cap B = \{2, 4\}$
(ii) $B \cap C = \{4, 5\}$
(iii) $A \cap (B \cup C) = A \cap \{2, 3, 4, 5, 6, 7, 8\} = \{2, 3, 4\}$
(iv) $A \cup (B \cap C) = A \cup \{4, 5\} = \{1, 2, 3, 4, 5\}$
(v) Verify $A \cup (B \cap C) = (A \cap B) \cup (A \cap C)$:
$LHS = \{1, 2, 3, 4, 5\}, \quad RHS = \{2, 4\} \cup \{3, 4\} = \{1, 2, 3, 4, 5\}.$
Verified.

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13. Cartesian products $A \times B$ and $B \times A$:

$A = \{1, 2, 3\}, B = \{a, b\}$

(i) $A \times B = \{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\}$
(ii) $B \times A = \{(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)\}$
(iii) Prove $A \times B \neq B \times A$:
The order of pairs matters. For example, $(1, a) \neq (a, 1)$. Thus, $A \times B \neq B \times A$.

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14. Find $(A \cup B)'$ and $A \cap B'$:

$A = \{1, 2, 3, 4\}, B = \{2, 4, 6, 8, 10\}, S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$

(i) $A \cup B = \{1, 2, 3, 4, 6, 8, 10\}$, so $(A \cup B)' = S - (A \cup B) = \{5, 7, 9\}$
(ii) $B' = S - B = \{1, 3, 5, 7, 9\}$, so $A \cap B' = \{1, 3\}$.

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15. Prove $A \times (B \cup C) = (A \times B) \cup (A \times C)$:

Let $A = \{1, 2\}, B = \{2, 3\}, C = \{3, 4\}$.
(i) $B \cup C = \{2, 3, 4\}$, so $A \times (B \cup C) = \{(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4)\}$.
(ii) $(A \times B) \cup (A \times C) = \{(1, 2), (1, 3), (2, 2), (2, 3)\} \cup \{(1, 3), (1, 4), (2, 3), (2, 4)\}$.
Both are equal, so the proof holds.

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Would you like further details on any of these?

5 Relative Questions:

1. How does complement differ from intersection in set theory?
2. Can you explain Cartesian products with more than two sets?
3. What is the significance of disjoint sets in mathematical applications?
4. How can set theory help solve real-world problems like survey analysis?
5. Can you generalize $A - (B \cup C)$ for arbitrary sets?

Tip: Always double-check set operations by expanding elements explicitly—this avoids common errors.