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.