STATEMENT-1 If `A cup B=A cup C, A cap B= A cap C, then B=C`
STATEMENT-2: `A-(B cup C)=(A-B) cup (A-C)`
STATEMENT-3: If `A cap (B' cap C')= A cap B cap C`.

Answer

To solve the given statements step by step, we will analyze each statement and determine whether they are true or false.

### Statement 1: If \( A \cup B = A \cup C \) and \( A \cap B = A \cap C \), then \( B = C \).

**Step 1:** Start with the given equations:
- \( A \cup B = A \cup C \)
- \( A \cap B = A \cap C \)

**Step 2:** From \( A \cup B = A \cup C \), we can deduce that any element in \( B \) that is not in \( A \) must also be in \( C \) and vice versa.

**Step 3:** Take the intersection of both sides with \( C \):
\[ (A \cup B) \cap C = (A \cup C) \cap C \]
Using the distributive property:
\[ (A \cap C) \cup (B \cap C) = (A \cap C) \cup (C \cap C) \]
Since \( C \cap C = C \), we can simplify:
\[ (A \cap C) \cup (B \cap C) = (A \cap C) \cup C \]

**Step 4:** Since \( A \cap B = A \cap C \), we can substitute:
\[ A \cap B = A \cap C \Rightarrow B \cap C = C \]

**Step 5:** Now, since \( A \cap B = A \cap C \) and \( A \cup B = A \cup C \), we conclude that \( B = C \).

**Conclusion for Statement 1:** True.

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### Statement 2: \( A - (B \cup C) = (A - B) \cup (A - C) \)

**Step 1:** Start with the left-hand side:
\[ A - (B \cup C) = \{ x \in A : x \notin (B \cup C) \} \]

**Step 2:** This means \( x \in A \) and \( x \notin B \) and \( x \notin C \).

**Step 3:** Now consider the right-hand side:
\[ (A - B) \cup (A - C) = \{ x \in A : x \notin B \} \cup \{ x \in A : x \notin C \} \]

**Step 4:** This means \( x \in A \) and either \( x \notin B \) or \( x \notin C \).

**Step 5:** The left-hand side requires \( x \notin B \) and \( x \notin C \), while the right-hand side allows for \( x \) to be in \( A \) but not necessarily in both \( B \) and \( C \).

**Conclusion for Statement 2:** False.

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### Statement 3: If \( A \cap (B' \cap C') = A \cap B \cap C \).

**Step 1:** Start with the left-hand side:
\[ A \cap (B' \cap C') = A \cap (U - B) \cap (U - C) \]
Where \( U \) is the universal set.

**Step 2:** This means \( x \in A \) and \( x \notin B \) and \( x \notin C \).

**Step 3:** The right-hand side \( A \cap B \cap C \) means \( x \in A \) and \( x \in B \) and \( x \in C \).

**Step 4:** Clearly, the left-hand side and right-hand side cannot be equal because the left-hand side requires \( x \) to be outside of both \( B \) and \( C \), while the right-hand side requires \( x \) to be inside both \( B \) and \( C \).

**Conclusion for Statement 3:** False.

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### Final Summary:
- **Statement 1:** True
- **Statement 2:** False
- **Statement 3:** False

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