Question

For two sets \(A, B\), if \(n(A)=15\), then what is \(n(B)\)? Statements: (I) \(n(A \cup B)=30\) (II) \(n(A \cap B)=8\)

• A. Statement (I) alone is sufficient to answer the question, but (II) alone is not sufficient.
• B. Statement (II) alone is sufficient to answer the question, but (I) alone is not sufficient.
• C. Both the statements (I) and (II) are sufficient to answer the question, but neither statement alone is sufficient.
• D. Both the statements (I) and (II) together are not sufficient to answer the question and additional data are required.

Hint:

For set-based data sufficiency problems, always use the union formula and check whether all unknowns can be determined.

Answer 1

The Correct Option is C

Concept: For two sets \(A\) and \(B\), the formula is: \[ n(A\cup B)=n(A)+n(B)-n(A\cap B) \] This formula helps us find unknown elements if sufficient data is available.

Step 1: Checking Statement (I).
Given: \[ n(A\cup B)=30 \] Also, \[ n(A)=15 \] Substituting: \[ 30=15+n(B)-n(A\cap B) \] Here, \(n(B)\) and \(n(A\cap B)\) are both unknown. So, Statement (I) alone is not sufficient.

Step 2: Checking Statement (II).
Given: \[ n(A\cap B)=8 \] But \(n(B)\) is still unknown and \(n(A\cup B)\) is unknown. So, Statement (II) alone is not sufficient.

Step 3: Checking both statements together.
Using: \[ n(A\cup B)=n(A)+n(B)-n(A\cap B) \] Substitute values: \[ 30=15+n(B)-8 \] \[ 30=7+n(B) \] \[ n(B)=23 \] Thus, \(n(B)\) can be uniquely determined. Hence, both statements together are sufficient.