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Document Content and Problems:

The document appears to be a chapter on set theory from a textbook, possibly for a university course, focusing on understanding and practice problems. Here's the breakdown:

Sets defined:

• $S = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$
• $A = \{2, 4, 5, 6, 8\}$
• $B = \{2, 3, 5, 7\}$
• $C = \{2, 3\}$

Problems listed:

1. Describe $S$
2. Describe set $C$ (complement)
3. Describe set $D$ if $D = \{2, 3, 4, 5, 6, 7, 9\}$
4. Operations: a. $A \cup B$ b. $A \cup B \cup C$
5. Given that $S$ is the universal set:
   • $A \subseteq S$
   • $B \subseteq S$
   • $C \subseteq S$
   • Describe $A \cap B \cup C$
   • Describe $A \cap B \cup C$
   • Describe $A \cap B \cup C$

Solutions:

1. $S$

   • $S$ is already defined as $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$.

2. $C$ Complement

   • $C' = S - C = \{1, 4, 5, 6, 7, 8, 9\}$

3. Set $D$

   • $D = \{2, 3, 4, 5, 6, 7, 9\}$

4. Operations a. $A \cup B = \{2, 3, 4, 5, 6, 7, 8\}$ b. $A \cup B \cup C = \{2, 3, 4, 5, 6, 7, 8\}$ (Note: $C$ is already included in $B$)

5. Intersection and Union

   • $A \cap B \cup C = (\{2, 4, 5, 6, 8\} \cap \{2, 3, 5, 7\}) \cup \{2, 3\} = \{2, 5\} \cup \{2, 3\} = \{2, 3, 5\}$

Each of these operations has been calculated based on set theory principles. Would you like any details on these operations, or do you have other questions about set theory?

Possible Further Questions:

1. How would you calculate the complement of $B$ in $S$?
2. What is the intersection of $A$, $B$, and $C$?
3. How does the symmetric difference between sets work?
4. Can you explain the difference between intersection and union in set theory?
5. How can the principles of set theory be applied to solve problems in other mathematical domains?

Tip:

Remember, in set theory, the union operation combines all unique elements from the sets involved, while the intersection operation identifies only the elements that are common to all sets involved.