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Statement-1 If Sets A and B have three a...

Statement-1 If Sets A and B have three and six elements respectively, then the minimum number of elements in `A uu B` is 6.
Statement-2 `AnnB=3`.

A

Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for Statement-1

B

Statement-1 is true, Statement-2 is true, Statement-2 is not a correct explanation for Statement-1

C

Statement-1 is true, Statement-2 is false

D

Statement-1 is false, Statement-2 is true

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the two statements given: **Statement 1:** If Sets A and B have three and six elements respectively, then the minimum number of elements in \( A \cup B \) is 6. **Statement 2:** \( A \cap B = 3 \). ### Step-by-step Solution: 1. **Understanding the Sets:** - Let the number of elements in Set A be \( |A| = 3 \). - Let the number of elements in Set B be \( |B| = 6 \). 2. **Using the Formula for Union of Sets:** - The formula for the number of elements in the union of two sets is given by: \[ |A \cup B| = |A| + |B| - |A \cap B| \] 3. **Identifying the Maximum Intersection:** - The maximum number of elements that can be in the intersection \( |A \cap B| \) cannot exceed the number of elements in the smaller set, which is Set A. Therefore, the maximum value of \( |A \cap B| \) is: \[ |A \cap B| \leq |A| = 3 \] 4. **Calculating the Minimum Union:** - To find the minimum number of elements in \( A \cup B \), we assume the maximum intersection: \[ |A \cup B| = |A| + |B| - |A \cap B| = 3 + 6 - 3 = 6 \] - Thus, the minimum number of elements in \( A \cup B \) is indeed 6. 5. **Evaluating Statement 2:** - Statement 2 claims that \( |A \cap B| = 3 \). This means that all elements of Set A are also in Set B. - If \( |A \cap B| = 3 \), then the calculation for \( |A \cup B| \) confirms that: \[ |A \cup B| = 3 + 6 - 3 = 6 \] - Therefore, Statement 2 is true and it supports Statement 1. ### Conclusion: - Both statements are true, and Statement 2 correctly explains Statement 1. ### Final Answer: - The correct option is: **Option A** - Statement 1 is true, Statement 2 is true, and Statement 2 is the correct explanation for Statement 1.

To solve the problem, we need to analyze the two statements given: **Statement 1:** If Sets A and B have three and six elements respectively, then the minimum number of elements in \( A \cup B \) is 6. **Statement 2:** \( A \cap B = 3 \). ### Step-by-step Solution: ...
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