Given U=x|x is an integer and 0 < x <= 20, A=5,7,9,11,13,17,19, B=2,4,8,10,12,18,20, C=3,9,12,15,18. Draw Venn diagrams and find the following sets: A n B n C, A' n B' n C, A' U B' U C', (A n B) n C, A U (B n C), (A n B) U C.

$$U = \{ n \in \mathbb{Z} \mid 0 \leq n \leq 20 \} = \{0, 1, 2, \dots, 20\}$$ $$A = \{5, 7, 9, 11, 17, 19\}$$ $$B = \{3, 4, 8, 10, 18, 20\}$$ $$C = \{2, 4, 6, 10, 18\}$$

Only $A$: $A \setminus (B \cup C) = \{5,7,9,11,17,19\}$

Only $B$: $B \setminus (A \cup C) = B \setminus C = \{3,8,20\}$

Only $C$: $C \setminus (A \cup B) = C \setminus B = \{2,6\}$

$A \cap B \setminus C = \emptyset$

$A \cap C \setminus B = \emptyset$

$B \cap C \setminus A = B \cap C = \{4,10,18\}$

$A \cap B \cap C = \emptyset$

Outside $A \cup B \cup C$: $U \setminus (A \cup B \cup C) = \{0,1,12,13,14,15,16\}$

Place these elements in the corresponding regions of the 3-circle Venn diagram inside $U$.

$A \cap B = \emptyset$ (no common elements).

Thus, $A \cap B \cap C' = \emptyset$.

Venn diagram: no regions shaded.

\emptyset

$A \cap B = \emptyset \implies A \subseteq B'$.

$A \cap C = \emptyset \implies A \subseteq C'$.

Thus, $A \cap B' \cap C' = A = \{5,7,9,11,17,19\}$.

Venn diagram: shade only $A$ region.

{5,7,9,11,17,19}

$A \cap B = \emptyset \implies B \subseteq A'$.

$B \cap C' = B \setminus C = \{3,8,20\}$.

Thus, $A' \cap B \cap C' = \{3,8,20\}$.

Venn diagram: shade only $B$ region.

{3,8,20}

$C \setminus (A \cup B) = \{2,6\}$.

$A \cup B \cup C' = U \setminus \{2,6\} = \{0,1,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20\}$.

Venn diagram: shade all regions except only $C$.

{0,1,3,4,5,7,8,9,10,11,12,13,14,15,16,17,18,19,20}