The expression \((B \cap C)'\) represents the complement of the intersection of sets \(B\) and \(C\).

### Understanding the Expression

1. **Intersection**: \(B \cap C\) is the set of all elements that are common to both sets \(B\) and \(C\).

2. **Complement**: The complement of a set, denoted by a prime (\('\)), consists of all elements that are not in that set, relative to a universal set \(U\).

### Applying These Definitions

- **Step 1**: Find the intersection \(B \cap C\).
- **Step 2**: Determine the complement \((B \cap C)'\), which includes all elements in the universal set \(U\) that are not in \(B \cap C\).

### Alternative Representation Using De Morgan's Laws

By De Morgan's Law:
\[
(B \cap C)' = B' \cup C'
\]

This means that the complement of the intersection of \(B\) and \(C\) is the union of their individual complements.

### Conclusion

- \((B \cap C)'\) includes all elements that are in \(B'\) (not in \(B\)) or in \(C'\) (not in \(C\)).

Would you like to see more details or have any specific questions about this?

### Related Questions:

1. What is the complement of a union of two sets?
2. How do De Morgan's laws apply to three or more sets?
3. Can you find the complement of an intersection using a Venn diagram?
4. How do you prove De Morgan's laws mathematically?
5. What is the symmetric difference between two sets?

**Tip**: When working with set operations, Venn diagrams are a helpful tool to visualize relationships and understand concepts like intersection and complement.

Explore how to find the complement of the intersection of sets B and C, using De Morgan's Laws. This set theory problem covers key concepts such as intersection, complement, and union, making it suitable for students in Grades 10-12.

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