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 $C$).

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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.