The given problem is:
The set $(A \cap B') \cup (B \cap C)$ is equal to...

Let’s analyze this step by step.

Expression:

$(A \cap B') \cup (B \cap C)$

• $A \cap B'$ means the intersection of A and the complement of B. This contains all elements in $A$ but not in $B$.
• $B \cap C$ means the intersection of B and C, i.e., all elements common to both $B$ and $C$.

Finally, we take the union of these two intersections, which means combining all elements from both sets.

Options:

• (A) $A \cup B \cup C$
• (B) $A' \cup B$
• (C) $A' \cup C$
• (D) $A' \cap B$

Correct Answer:

The correct answer is (B) $A' \cup B$.

Reason:
$(A \cap B') \cup (B \cap C)$ simplifies to a set that contains all elements not in $A$ (i.e., $A'$) or elements in $B$. This is because:

1. $A \cap B'$ contributes elements in A but not in B.
2. $B \cap C$ contributes elements common to B and C.

Thus, the entire union is equivalent to the set $A' \cup B$.

Let me know if you need further details! Here are five related questions for better understanding:

1. What is the complement of a set in set theory?
2. How do intersection and union operations work?
3. Can you give an example of De Morgan’s laws?
4. What happens when we take the union of a set with its complement?
5. How do Venn diagrams help in visualizing set operations?

Tip: Always double-check your set operations against known set identities—especially when working with complements and intersections.