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The problem asks to solve set operations involving the universal set \( U \) and the subsets \( A \), \( B \), and \( C \), as well as to draw Venn diagrams.

Here are the given sets:

\[
U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14\}
\]
\[
A = \{1, 2, 3, 4, 5, 6\}, \quad B = \{3, 4, 6, 7, 13\}, \quad C = \{5, 6, 7, 8, 9, 10, 11, 12, 13\}
\]

You are asked to perform the following operations:

### a. \( A \cap B \)
This is the intersection of sets \( A \) and \( B \) (common elements between \( A \) and \( B \)).

\[
A \cap B = \{3, 4, 6\}
\]

### b. \( A \cup B \)
This is the union of sets \( A \) and \( B \) (all elements in \( A \) or \( B \)).

\[
A \cup B = \{1, 2, 3, 4, 5, 6, 7, 13\}
\]

### c. \( C \cap A \)
This is the intersection of sets \( C \) and \( A \).

\[
C \cap A = \{5, 6\}
\]

### d. \( B \cap C \)
This is the intersection of sets \( B \) and \( C \).

\[
B \cap C = \{6, 7, 13\}
\]

### e. \( A \cup B \cup C \)
This is the union of sets \( A \), \( B \), and \( C \) (all elements in \( A \), \( B \), or \( C \)).

\[
A \cup B \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}
\]

### f. \( A \cap B \cap C \)
This is the intersection of sets \( A \), \( B \), and \( C \).

\[
A \cap B \cap C = \{6\}
\]

### g. \( (A \cup B) \cap C \)
First, calculate \( A \cup B \), and then find the intersection with \( C \).

\[
A \cup B = \{1, 2, 3, 4, 5, 6, 7, 13\}
\]
\[
(A \cup B) \cap C = \{5, 6, 7, 13\}
\]

### h. \( \overline{A} \cap \overline{B} \cap C \)
Here, \( \overline{A} \) and \( \overline{B} \) are the complements of \( A \) and \( B \) with respect to \( U \).

\[
\overline{A} = U - A = \{7, 8, 9, 10, 11, 12, 13, 14\}
\]
\[
\overline{B} = U - B = \{1, 2, 5, 8, 9, 10, 11, 12, 14\}
\]
\[
\overline{A} \cap \overline{B} = \{8, 9, 10, 11, 12, 14\}
\]
\[
\overline{A} \cap \overline{B} \cap C = \{8, 9, 10, 11, 12\}
\]

### i. \( A \cap B \cap \overline{C} \)
First, find \( \overline{C} \) (complement of \( C \)).

\[
\overline{C} = U - C = \{1, 2, 3, 4, 14\}
\]
\[
A \cap B = \{3, 4, 6\}
\]
\[
A \cap B \cap \overline{C} = \{3, 4\}
\]

Let me know if you want the details explained or have any questions! Here are five related questions for further practice:

1. What is \( \overline{A \cup B} \)?
2. Find \( \overline{B \cap C} \).
3. Calculate \( A \cup (B \cap C) \).
4. What is \( (A \cap \overline{B}) \cup C \)?
5. Find \( \overline{A \cap B \cap C} \).

**Tip:** When dealing with set complements, remember they represent everything in the universal set that is **not** in the given set.

Misalkan diketahui:
U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14}
A = {1,2,3,4,5,6}
B = {3,4,6,7,13}
C = {5,6,7,8,9,10,11,12,13}
Gambarkan diagram Venn-nya kemudian selesaikan:
a. A ∩ B
b. A ∪ B
c. C ∩ A
d. B ∩ C
e. A ∪ B ∪ C
f. A ∩ B ∩ C
g. (A ∪ B) ∩ C
h. (A ∩ B ∩ C)^C
i. A ∩ B ∩ C^C

Explore a detailed set theory problem involving the universal set and subsets A, B, and C. Learn to solve intersections, unions, and complements using set theory operations, including Venn diagrams. This problem is designed for students in Grades 9-12.

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