The cardinality of a finite set is the number of elements in the set. What is the cardinality of set A ? 2 is the cardinality of exactly 6 subsets of set A . Set A has a total of 16 subsets, including the empty set and set A itself.

The cardinality of a finite set is the number of elements in the set. What is the cardinality of set...

GMAT Data Sufficiency : (DS) Questions

Source: Official Guide

Data Sufficiency

DS - Sets and Probability

HARD

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Notes

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The cardinality of a finite set is the number of elements in the set. What is the cardinality of set \(\mathrm{A}\)?

\(2\) is the cardinality of exactly \(6\) subsets of set \(\mathrm{A}\).
Set \(\mathrm{A}\) has a total of \(16\) subsets, including the empty set and set \(\mathrm{A}\) itself.

Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

Solution

markdown
## Understanding the Question

We need to find the **cardinality of set A** - that is, the exact number of elements in set A.

The question helpfully defines cardinality as the number of elements in a finite set. To answer this question, we need information that allows us to determine exactly one value for the number of elements in set A. If we can narrow it down to a single number, we have sufficiency.

### Key Insight
For any finite set with n elements:
- The total number of subsets = 2^n
- The number of subsets with exactly k elements = C(n,k) = n!/(k!(n-k)!)

These fundamental relationships from set theory will guide our analysis.

## Analyzing Statement 1

**Statement 1 tells us**: There are exactly 6 subsets of set A that contain exactly 2 elements.

If set A has n elements, then the number of 2-element subsets equals C(n,2) = n(n-1)/2.

Since we know this equals 6:
- n(n-1)/2 = 6
- n(n-1) = 12
- n² - n - 12 = 0
- (n-4)(n+3) = 0

This gives us n = 4 or n = -3. Since the number of elements must be positive, we have n = 4.

Let's verify: C(4,2) = 4×3/2 = 6 ✓

**[STOP - Sufficient!]** Statement 1 uniquely determines that set A has exactly 4 elements.

Statement 1 is **SUFFICIENT**.

## Analyzing Statement 2

**Now let's forget Statement 1 completely and analyze Statement 2 independently.**

**Statement 2 tells us**: Set A has a total of 16 subsets (including the empty set and set A itself).

For a set with n elements, the total number of subsets equals 2^n.

Since we know this equals 16:
- 2^n = 16
- 2^n = 2^4
- Therefore, n = 4

This is immediate when we recognize that 16 = 2^4. (Quick check: 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16)

**[STOP - Sufficient!]** Statement 2 uniquely determines that set A has exactly 4 elements.

Statement 2 is **SUFFICIENT**.

## The Answer: D

Both statements independently lead us to the same unique conclusion: set A has exactly 4 elements. Since each statement alone is sufficient to answer the question, the answer is D.

**Answer Choice D**: "Each statement alone is sufficient."

Answer Choices Explained

Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

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markdown ## Understanding the Question We need to find the **cardinality of set A** - that is, the exact number of elements in set A. The question helpfully defines cardinality as the number of elements in a finite set. To answer this question, we need information that allows us to determine exactly one value for the number of elements in set A. If we can narrow it down to a single number, we have sufficiency. ### Key Insight For any finite set with n elements: - The total number of subsets = 2^n - The number of subsets with exactly k elements = C(n,k) = n!/(k!(n-k)!) These fundamental relationships from set theory will guide our analysis. ## Analyzing Statement 1 **Statement 1 tells us**: There are exactly 6 subsets of set A that contain exactly 2 elements. If set A has n elements, then the number of 2-element subsets equals C(n,2) = n(n-1)/2. Since we know this equals 6: - n(n-1)/2 = 6 - n(n-1) = 12 - n² - n - 12 = 0 - (n-4)(n+3) = 0 This gives us n = 4 or n = -3. Since the number of elements must be positive, we have n = 4. Let's verify: C(4,2) = 4×3/2 = 6 ✓ **[STOP - Sufficient!]** Statement 1 uniquely determines that set A has exactly 4 elements. Statement 1 is **SUFFICIENT**. ## Analyzing Statement 2 **Now let's forget Statement 1 completely and analyze Statement 2 independently.** **Statement 2 tells us**: Set A has a total of 16 subsets (including the empty set and set A itself). For a set with n elements, the total number of subsets equals 2^n. Since we know this equals 16: - 2^n = 16 - 2^n = 2^4 - Therefore, n = 4 This is immediate when we recognize that 16 = 2^4. (Quick check: 2^1 = 2, 2^2 = 4, 2^3 = 8, 2^4 = 16) **[STOP - Sufficient!]** Statement 2 uniquely determines that set A has exactly 4 elements. Statement 2 is **SUFFICIENT**. ## The Answer: D Both statements independently lead us to the same unique conclusion: set A has exactly 4 elements. Since each statement alone is sufficient to answer the question, the answer is D. **Answer Choice D**: "Each statement alone is sufficient."