Let's analyze the given sets:
\( A = \{1, 2, 3, 4, 5, 8, 9, 12, 15, 17\} \)
\( B = \{1, 2, 3, 4, 8, 9, 10\} \)
\( C = \{1, 2, 7, 9, 10, 11, 13\} \)
Now, let's find the requested sets:
1. \( A \cup B \cap C \): This is the union of set A and the intersection of sets B and C.
\( A \cup B = \{1, 2, 3, 4, 5, 8, 9, 10, 12, 15, 17\} \)
\( A \cup B \cap C = A \cup (B \cap C) = \{1, 2, 3, 4, 5, 8, 9, 10, 12, 15, 17\} \)
2. \( A \cap \sim B \cup C \): This is the intersection of set A and the union of the complement of B and C.
\( \sim B \) is the complement of B with respect to a universal set U. Since the universal set is not specified, let's assume it includes all the numbers mentioned in any of the sets. So, \( \sim B = \{5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17\} \).
\( A \cap \sim B = \{5, 12, 15, 17\} \)
\( A \cap \sim B \cup C = \{1, 2, 5, 7, 9, 10, 11, 12, 13, 15, 17\} \)
3. \( A \cap B \cup C \): This is the intersection of set A and the union of sets B and C.
\( A \cap B = \{1, 2, 3, 4, 8, 9\} \)
\( A \cap B \cup C = \{1, 2, 3, 4, 8, 9, 7, 10, 11, 13\} \)
4. \( A \cap \sim C \): This is the intersection of set A and the complement of C.
\( \sim C = \{3, 4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 17\} \)
\( A \cap \sim C = \{3, 4, 5, 8, 9, 12, 15, 17\} \)
These are the results for the specified set operations.
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