Let U = 1, 2, ..., 20. Define sets A, B, and C within U. Draw a Venn diagram for A, B, C. A = squares, B = factors of 20, C = multiples of 4.

Square numbers are numbers that are the product of an integer multiplied by itself.

$$A = \{x \in U \mid x = n^2 \text{ for some integer } n\}$$

The square numbers in $U$ are:
$1^2 = 1$
$2^2 = 4$
$3^2 = 9$
$4^2 = 16$
$5^2 = 25$ (This is greater than 20, so it's not in $U$)

$$A = \{1, 4, 9, 16\}$$

Factors of 20 are numbers that divide 20 evenly.

$$B = \{x \in U \mid 20 \div x \text{ is an integer}\}$$

The factors of 20 are:
$1, 2, 4, 5, 10, 20$

$$B = \{1, 2, 4, 5, 10, 20\}$$

Multiples of 4 are numbers that can be divided by 4 evenly.

$$C = \{x \in U \mid x = 4n \text{ for some integer } n\}$$

The multiples of 4 in $U$ are:
$4 \times 1 = 4$
$4 \times 2 = 8$
$4 \times 3 = 12$
$4 \times 4 = 16$
$4 \times 5 = 20$
$4 \times 6 = 24$ (This is greater than 20, so it's not in $U$)

$$C = \{4, 8, 12, 16, 20\}$$

• $A \cap B$: Elements common to A and B. $A = \{1, 4, 9, 16\}$ $B = \{1, 2, 4, 5, 10, 20\}$

$$A \cap B = \{1, 4\}$$

• $A \cap C$: Elements common to A and C. $A = \{1, 4, 9, 16\}$ $C = \{4, 8, 12, 16, 20\}$

$$A \cap C = \{4, 16\}$$

• $B \cap C$: Elements common to B and C. $B = \{1, 2, 4, 5, 10, 20\}$ $C = \{4, 8, 12, 16, 20\}$

$$B \cap C = \{4, 20\}$$

• $A \cap B \cap C$: Elements common to A, B, and C. From $A \cap B = \{1, 4\}$ and $C = \{4, 8, 12, 16, 20\}$, the common element is 4.

$$A \cap B \cap C = \{4\}$$

• Elements in $A \cap B \cap C$: $\{4\}$
• Elements in $A \cap B$ only (not in C): $\{1\}$
• Elements in $A \cap C$ only (not in B): $\{16\}$
• Elements in $B \cap C$ only (not in A): $\{20\}$
• Elements in A only: $A \setminus (A \cap B \cup A \cap C) = \{1, 4, 9, 16\} \setminus \{1, 4, 16\} = \{9\}$
• Elements in B only: $B \setminus (A \cap B \cup B \cap C) = \{1, 2, 4, 5, 10, 20\} \setminus \{1, 4, 20\} = \{2, 5, 10\}$
• Elements in C only: $C \setminus (A \cap C \cup B \cap C) = \{4, 8, 12, 16, 20\} \setminus \{4, 16, 20\} = \{8, 12\}$
• Elements outside A, B, and C (in $U$ but not in $A \cup B \cup C$): $A \cup B \cup C = \{1, 2, 4, 5, 8, 9, 10, 12, 16, 20\}$ $U \setminus (A \cup B \cup C) = \{3, 6, 7, 11, 13, 14, 15, 17, 18, 19\}$

A Venn diagram would consist of a rectangle representing the universal set $U$, and three overlapping circles representing sets A, B, and C. The elements would be placed in the corresponding regions as follows:

• Region $A \cap B \cap C$: $\{4\}$
• Region $(A \cap B) \setminus C$: $\{1\}$
• Region $(A \cap C) \setminus B$: $\{16\}$
• Region $(B \cap C) \setminus A$: $\{20\}$
• Region $A \setminus (B \cup C)$: $\{9\}$
• Region $B \setminus (A \cup C)$: $\{2, 5, 10\}$
• Region $C \setminus (A \cup B)$: $\{8, 12\}$
• Region $U \setminus (A \cup B \cup C)$ (outside all circles): $\{3, 6, 7, 11, 13, 14, 15, 17, 18, 19\}$