1. List the elements in $\alpha$, $B$, and $U$: - $\alpha = \{1, 9\}$ - $B = \{21, 4, 6, 8, 10\}$ - $U = \{0, 23, 31\}$ 2. List the elements in $\alpha$, $B'$, and $U'$: - $B'$ is the complement of $B$ relative to the universal set $U$. Since $U = \{0, 23, 31\}$ and $B = \{21, 4, 6, 8, 10\}$, none of $B$'s elements are in $U$, so $B' = U = \{0, 23, 31\}$ - $U'$ is the complement of $U$ relative to itself, so $U' = \emptyset$ - $\alpha = \{1, 9\}$ (unchanged) 3. Find: a) $\alpha \cup B \cup \theta$ - Given $\theta$ is the set of elements inside the Venn diagram region: $\{29, 13, 2, 3, 7, 5, 11, 17, 19, 15, 18\}$ - $\alpha \cup B \cup \theta = \{1, 9\} \cup \{21, 4, 6, 8, 10\} \cup \{29, 13, 2, 3, 7, 5, 11, 17, 19, 15, 18\}$ - Combine all unique elements: $$\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 18, 19, 21, 29\}$$ b) $\alpha' \cap B' \cap \theta'$ - $\alpha'$, complement of $\alpha$ relative to $U \cup B \cup \theta$ (all elements considered): - Total elements in $U \cup B \cup \theta \cup \alpha$ are $\{0,1,2,3,4,5,6,7,8,9,10,11,13,15,17,18,19,21,23,29,31\}$ - $\alpha = \{1,9\}$ so $\alpha' = \{0,2,3,4,5,6,7,8,10,11,13,15,17,18,19,21,23,29,31\}$ - $B' = \{0,23,31\}$ (from step 2) - $\theta'$, complement of $\theta$ relative to the total elements above: - $\theta = \{29, 13, 2, 3, 7, 5, 11, 17, 19, 15, 18\}$ - So $\theta' = \{0,1,4,6,8,9,10,21,23,31\}$ - Now find intersection: $$\alpha' \cap B' \cap \theta' = \{0,2,3,4,5,6,7,8,10,11,13,15,17,18,19,21,23,29,31\} \cap \{0,23,31\} \cap \{0,1,4,6,8,9,10,21,23,31\}$$ - First intersect $\alpha'$ and $B'$: $$\{0,23,31\}$$ - Then intersect with $\theta'$: $$\{0,23,31\} \cap \{0,1,4,6,8,9,10,21,23,31\} = \{0,23,31\}$$ - So $\alpha' \cap B' \cap \theta' = \{0, 23, 31\}$ c) $\alpha \cap B \cap \theta$ - $\alpha = \{1,9\}$, $B = \{21,4,6,8,10\}$, $\theta = \{29,13,2,3,7,5,11,17,19,15,18\}$ - Intersection of these three sets is empty because no element is common in all three: $$\alpha \cap B \cap \theta = \emptyset$$ d) $\alpha' \cup B' \cup \theta'$ - From above: - $\alpha' = \{0,2,3,4,5,6,7,8,10,11,13,15,17,18,19,21,23,29,31\}$ - $B' = \{0,23,31\}$ - $\theta' = \{0,1,4,6,8,9,10,21,23,31\}$ - Union all: $$\{0,1,2,3,4,5,6,7,8,9,10,11,13,15,17,18,19,21,23,29,31\}$$ 4. Compare results of 3a and 3b: - 3a: $\{1,2,3,4,5,6,7,8,9,10,11,13,15,17,18,19,21,29\}$ - 3b: $\{0,23,31\}$ - These two sets are complements relative to the universal set of all elements considered. - Conclusion: $\alpha \cup B \cup \theta$ and $\alpha' \cap B' \cap \theta'$ are complements. 5. The conclusion in 4 is restated: The union of sets $\alpha$, $B$, and $\theta$ is the complement of the intersection of their complements. This illustrates De Morgan's Law: $$ (\alpha \cup B \cup \theta)' = \alpha' \cap B' \cap \theta' $$ This confirms the relationship between union and intersection of complements in set theory.