# GMAT Math : DSQ: Understanding arithmetic sets

## Example Questions

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### Example Question #11 : Sets

Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.

If real number  were to be placed in its correct region in the diagram, which one would it be - I, II, III, IV, or V?

Statement 1: If , then  would be placed in Region IV.

Statement 2: If , then  would be placed in Region IV.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Region IV comprises the rational numbers that are not integers. A number is rational if and only if it can be expressed as the quotient of integers.

From Statement 1 alone, it can be inferred that  is rational, and that it is not an integer. Since , it follows that . However, this is not sufficient to narrow it down completely.

For example:

If , then , a natural number, putting it in Region I.

If , then , a rational number but not an integer, putting it in Region IV.

From Statement 2 alone, it can be inferred that  is rational, and that it is not an integer. From , it follows that . The nonzero rational numbers are closed under division, so  must be a rational number. However, since  is not an integer,  cannot be an integer, since the integers are closed under multiplication. Therefore, Statement 2 alone proves that  belongs in Region IV.

### Example Question #12 : Sets

Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.

If real number  were to be placed in its correct region in the diagram, which one would it be - I, II, III, IV, or V?

Statement 1: If , then  would be placed in Region I.

Statement 2: If , then  would be placed in Region I.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

Region I comprises the natural numbers -

From Statement 1 alone,  is a natural number; since , it follows that  is the difference of a natural number and 7 - that is,

could be in any of three regions - I, II, or III.

Conversely, from Statement 2 alone,   is the sum of a natural number and 7 - that is,

must be a natural number and it must be in Region I.

### Example Question #13 : Sets

Examine the above diagram, which shows a Venn diagram representing the sets of real numbers.

If real number  were to be placed in its correct region in the diagram, which one would it be - I, II, III, IV, or V?

Statement 1: If , then  would be in Region I.

Statement 2: If , then  would be in Region III.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. It cannot be determined what region  is in.

For example, suppose , which is in Region I (the set of natural numbers, or positive integers). It is possible that , putting it in Region I, or , putting it in Region III (the set of integers that are not whole numbers - that is, the set of negative integers).

Assume Statement 2 alone. It cannot be determined what region  is in.

For example, suppose , which is in Region III; then , which is also in Region III. But suppose ; then , which, as an irrational number, is in Region V.

Now assume both statements. Then  has an integer as a square and an integer as a cube.  must either be an integer or an irrational number. But

, making it the quotient of integers, which is rational. Therefore,  is an integer. Furthermore, its cube is negative, so  is negative. The two statements together prove that  is a negative integer, which belongs in Region III.

### Example Question #11 : Sets

How many elements are in set ?

Statement 1:  has exactly  subsets.

Statement 2:  has exactly  proper subsets.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

A set with  elements has exactly  subsets in all, and  proper subsets (every subset except one - the set itself).

From Statement 1, since  has  subsets, it follows that it has 6 elements. From Statement 2, since  has 63 proper subsets, it has 64 subsets total, and, again, 6 elements. Either statement alone is sufficient.

### Example Question #15 : Sets

Which, if either, is the greater number:  or  ?

Statement 1:

Statement 2:

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 alone gives insufficient information. For example, if , then:

or

Since , it is unclear which of  and  is greater, if either.

Statement 2 gives insufficient information; if  is positive,  is negative, and vice versa.

Assume both to be true. The two statements form a system of equations that can be solved using substitution:

Case 1:

Case 2:

This equation has no solution.

Therefore, the only possible solution is . Therefore, it can be concluded that .

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