It seems the notation you've used might have some formatting errors or typographical issues. Let's try to organize your problem in standard set notation to clarify the operations. From what I understand, the given sets are:

• $\Omega = \{1,2,3,4,5,6,7,8,9,10\}$ (Universal set)
• $A = \{2,4,6,8,10\}$
• $B = \{1,3,5,7,9\}$
• $C = \{1,4,9\}$
• $D = \{1,8\}$
• $E = \{1,2,3,5,7\}$

Based on this, I'll try to find the expressions you're asking for. Let's proceed:

1. $A \cap B$ (Intersection of $A$ and $B$):

   • $A = \{2,4,6,8,10\}$
   • $B = \{1,3,5,7,9\}$
   • $A \cap B = \emptyset$ (No common elements)

2. $B \cup C$ (Union of $B$ and $C$):

   • $B = \{1,3,5,7,9\}$
   • $C = \{1,4,9\}$
   • $B \cup C = \{1,3,4,5,7,9\}$

3. $(A \cap B)'$ (Complement of $A \cap B$):

   • $A \cap B = \emptyset$, so:
   • $(A \cap B)' = \Omega = \{1,2,3,4,5,6,7,8,9,10\}$

4. $A \cap C$ (Intersection of $A$ and $C$):

   • $A = \{2,4,6,8,10\}$
   • $C = \{1,4,9\}$
   • $A \cap C = \{4\}$

5. $B \cap D$ (Intersection of $B$ and $D$):

   • $B = \{1,3,5,7,9\}$
   • $D = \{1,8\}$
   • $B \cap D = \{1\}$

6. $B' \cap C$ (Intersection of complement of $B$ and $C$):

   • $B' = \Omega - B = \{2,4,6,8,10\}$
   • $C = \{1,4,9\}$
   • $B' \cap C = \{4\}$

7. $(A \cap C)'$ (Complement of $A \cap C$):

   • $A \cap C = \{4\}$
   • $(A \cap C)' = \Omega - \{4\} = \{1,2,3,5,6,7,8,9,10\}$

8. $A \cap D$ (Intersection of $A$ and $D$):

   • $A = \{2,4,6,8,10\}$
   • $D = \{1,8\}$
   • $A \cap D = \{8\}$

9. $B \cap E$ (Intersection of $B$ and $E$):

   • $B = \{1,3,5,7,9\}$
   • $E = \{1,2,3,5,7\}$
   • $B \cap E = \{1,3,5,7\}$

10. $B \cap C$ (Intersection of $B$ and $C$):

    • $B = \{1,3,5,7,9\}$
    • $C = \{1,4,9\}$
    • $B \cap C = \{1,9\}$

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I hope this helps clarify things! Let me know if you'd like any more specific operations worked through or if you'd like further details.

5 Related Questions

1. How do you find the union and intersection of two sets?
2. What is the significance of the universal set in set theory?
3. How does complement operation work in set theory?
4. What are De Morgan's Laws in set theory?
5. How are Venn diagrams used to represent sets?

Tip: In set operations, intersections give the common elements, and unions combine all unique elements from both sets.