STATEMENT-1 Number of elements belonging to exactly, 2q, the sets of A,B,C is `n(A cap B)+n(B cap C)+n(C cap B)-3n(A cap B capC)`
STATEMENT-2 Number of elements belonging to exactly one of the sets A,B and C is `n(A cup B cup C)-n(A cap B)-n(A cap C)+2n(A cap B capC)`

Answer

To solve the problem, we will analyze both statements step by step.

### Statement 1:
**Number of elements belonging to exactly 2 of the sets A, B, C is given by:**
\[ n(A \cap B) + n(B \cap C) + n(C \cap A) - 3n(A \cap B \cap C) \]

#### Step 1: Understand the Venn Diagram
- Draw a Venn diagram with three circles representing sets A, B, and C.
- Identify the regions where elements belong to exactly two sets.

#### Step 2: Identify the Regions
- Let:
- Region 1: Elements in \( A \cap B \) but not in \( C \)
- Region 2: Elements in \( B \cap C \) but not in \( A \)
- Region 3: Elements in \( C \cap A \) but not in \( B \)
- The middle region (where all three sets overlap) is \( A \cap B \cap C \).

#### Step 3: Write the Expressions for Each Region
- For Region 1: \( n(A \cap B) - n(A \cap B \cap C) \)
- For Region 2: \( n(B \cap C) - n(A \cap B \cap C) \)
- For Region 3: \( n(C \cap A) - n(A \cap B \cap C) \)

#### Step 4: Combine the Expressions
- Total number of elements belonging to exactly two sets:
\[ (n(A \cap B) - n(A \cap B \cap C)) + (n(B \cap C) - n(A \cap B \cap C)) + (n(C \cap A) - n(A \cap B \cap C)) \]
- This simplifies to:
\[ n(A \cap B) + n(B \cap C) + n(C \cap A) - 3n(A \cap B \cap C) \]

### Conclusion for Statement 1
- The statement is true.

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### Statement 2:
**Number of elements belonging to exactly one of the sets A, B, and C is given by:**
\[ n(A \cup B \cup C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + 2n(A \cap B \cap C) \]

#### Step 1: Understand the Venn Diagram
- Again, draw a Venn diagram with three circles representing sets A, B, and C.
- Identify the regions where elements belong to exactly one set.

#### Step 2: Identify the Regions
- Let:
- Region 1: Elements only in A
- Region 2: Elements only in B
- Region 3: Elements only in C

#### Step 3: Write the Expressions for Each Region
- For Region 1: \( n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C) \)
- For Region 2: \( n(B) - n(A \cap B) - n(B \cap C) + n(A \cap B \cap C) \)
- For Region 3: \( n(C) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \)

#### Step 4: Combine the Expressions
- Total number of elements belonging to exactly one set:
\[ (n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C)) + (n(B) - n(A \cap B) - n(B \cap C) + n(A \cap B \cap C)) + (n(C) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)) \]
- This simplifies to:
\[ n(A) + n(B) + n(C) - (n(A \cap B) + n(A \cap C) + n(B \cap C)) + 3n(A \cap B \cap C) \]

### Conclusion for Statement 2
- The statement is false as it does not correctly represent the number of elements belonging to exactly one of the sets.

### Final Conclusion
- **Statement 1 is True.**
- **Statement 2 is False.**

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