Given a universal set 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 to show the universal set U and the subsets A, B, C. Find the following sets: (a) A ∩ B ∩ C (b) A' ∩ B' ∩ C' (c) A' ∩ B' ∩ C (d) (A ∩ B) ∩ C (e) A U (B ∩ C) (f) (A ∩ B) U C

Step 1: List the given sets.

(a) Venn diagram regions for $A$, $B$, $C$

Step 2: Only $A$ (in $A$, not $B$, not $C$):
$A \setminus (B \cup C) = \{5,7,9,11,17,19\} \setminus (\{2,5,8,9,10,12,17,18\} \cup \{3,9,12,15,18\})$
$= \{5,7,9,11,17,19\} \setminus \{2,3,5,8,9,10,12,15,17,18\}$
$= \{7, 11, 19\}$

Step 3: Only $B$ (in $B$, not $A$, not $C$):
$B \setminus (A \cup C) = \{2,5,8,9,10,12,17,18\} \setminus (\{5,7,9,11,17,19\} \cup \{3,9,12,15,18\})$
$= \{2,5,8,9,10,12,17,18\} \setminus \{3,5,7,9,11,12,15,17,18,19\}$
$= \{2, 8, 10\}$

Step 4: Only $C$ (in $C$, not $A$, not $B$):
$C \setminus (A \cup B) = \{3,9,12,15,18\} \setminus (\{5,7,9,11,17,19\} \cup \{2,5,8,9,10,12,17,18\})$
$= \{3,9,12,15,18\} \setminus \{2,5,7,8,9,10,11,12,17,18,19\}$
$= \{3, 15\}$

Step 5: $A \cap B$ only (not $C$):
$(A \cap B) \setminus C = (\{5,9,17\}) \setminus \{3,9,12,15,18\} = \{5, 17\}$

Step 6: $A \cap C$ only (not $B$):
$(A \cap C) \setminus B = (\{9\}) \setminus \{2,5,8,9,10,12,17,18\} = \emptyset$

Step 7: $B \cap C$ only (not $A$):
$(B \cap C) \setminus A = (\{9,12,18\}) \setminus \{5,7,9,11,17,19\} = \{12, 18\}$

Step 8: $A \cap B \cap C$:
$A \cap B \cap C = \{5,7,9,11,17,19\} \cap \{2,5,8,9,10,12,17,18\} \cap \{3,9,12,15,18\} = \{9\}$

Venn diagram summary:

• Only $A$: $\{7,11,19\}$
• Only $B$: $\{2,8,10\}$
• Only $C$: $\{3,15\}$
• $A\cap B$ only: $\{5,17\}$
• $A\cap C$ only: $\emptyset$
• $B\cap C$ only: $\{12,18\}$
• $A\cap B\cap C$: $\{9\}$
• Outside $A\cup B\cup C$: $\{U \setminus \{2,3,5,7,8,9,10,11,12,15,17,18,19\}\} = \{1,4,6,13,14,16\}$

(b) $A \cap B \cap C$
From Step 8: $\{9\}$
$\{9\}$

(c) $A \cup B \cup C$

Step 9: $A \cup B = \{2,5,7,8,9,10,11,12,17,18,19\}$
$A \cup B \cup C = \{2,5,7,8,9,10,11,12,17,18,19\} \cup \{3,9,12,15,18\} = \{2,3,5,7,8,9,10,11,12,15,17,18,19\}$
$\{2,3,5,7,8,9,10,11,12,15,17,18,19\}$

Step 1: List the given sets.

(i) $W = (A \cup B) \cup (C \cup D)$

Step 2: $A \cup B = \{1,2,3,5,6,7,8,9,10\}$
$C \cup D = \{3,5,6,7,8,9,10\}$
$W = \{1,2,3,5,6,7,8,9,10\} \cup \{3,5,6,7,8,9,10\} = \{1,2,3,5,6,7,8,9,10\}$
$\{1,2,3,5,6,7,8,9,10\}$

(ii) $V = \{1,2,3,4,5\} \cup (B \cap C) \cup (A \cap D) \cup (B \cap D)$

Step 3: $B \cap C = \{2,5,6,9,10\} \cap \{3,5,6,7\} = \{5,6\}$

Step 4: $A \cap D = \{1,2,3,5,7,8\} \cap \{6,7,8,9,10\} = \{7,8\}$

Step 5: $B \cap D = \{2,5,6,9,10\} \cap \{6,7,8,9,10\} = \{6,9,10\}$

Step 6: $\{1,2,3,4,5\} \cup \{5,6\} \cup \{7,8\} \cup \{6,9,10\}$
$= \{1,2,3,4,5,6,7,8,9,10\}$
$\{1,2,3,4,5,6,7,8,9,10\}$