The Set S of Numbers has the following Properties GMAT Data Sufficiency

Question: The Set S of Numbers has the following Properties:

1.  If x is in S, then 1/x is in S.
2.  If both x and y are in S, then so is x + y.

Is 3 in S?

1.  1/3 is in S.
2. 1 is in S.

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

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

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

(D) EACH statement ALONE is sufficient.        

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

“The Set S of Numbers has the following Properties GMAT Data Sufficiency” - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation: 

Approach Solution 1:

The given statement for the question identifies that the set of S numbers has different properties. These properties need to be evaluated and identified by evaluating the two given statements to be sufficient or not. However, more than equation based evaluation, the answer requires to be evaluated based on a detailed understanding of the question.
Accordingly, both the statements are required to be proved and evaluated as follows below.
In order to find if 3 is in S or not, the following needs to be explained:
The first statement evaluates that ⅓ is in S.
This can be considered as ⅓ is in S. This further implies that \(\frac{1}{\frac1 3}\)helps in giving the answer 3.

Hence, the first statement for \(\frac{1}{x}\)being in S is absolutely correct.

The second statement states that 1 is in S. Accordingly, in order to prove whether this statement is sufficient or not, the following evaluations are to be considered.
This implies that, if 1 is in S, then considering S as x, it can be evaluated as-
x = S

then , \(\frac{1}{x}\) is in S. while x can be stated as 1, then the value may be 1.

Considering that the second statement evaluates having both x and y in S, it can be evaluated that both x and y are 1. Further , based on the second part of the second statement, it states that x + y is also in S. This further implies as follows:
x + y = 1 + 1
This equals 2.
Hence, this implies that 2 must be in S.
Further, if the x + y is equal to having properties proving that 2 is in S, it is also true that x + y may also equal 1 + 2 which is 3.
Hence, this statement is also sufficient.
Thus, each statement alone is sufficient to answer the question. Thus, the correct answer is D.

Correct Answer: D.

Approach Solution 2:

The wording is a little bit strange but anyway, we will be focussing on the options:

Option 1 –

1/3 is in S

--> According to (i) 1/(1/3)=3 must also be in S.

Hence, it is Sufficient.

Option 2 –

1 is in S --> according to (i) 1/1=1 (another 1) must also be in S

This states that according to option 2 :1+1=2 must also be in S --> 1+2=3 must also be in S.

Hence, this statement is also Sufficient.

Since both are sufficient, D is the correct answer.

Correct Answer: D.

Approach Solution 3:

Technically in a set we can't have repeated values.

So here, when we use Statement 2, we don't need to use the property that "If x is in S, then 1/x is in S" to prove that we have another '1' in the set -- you already know '1' is in the set, and there can only be one '1' in a set.

We can only come to the above solution since a set cannot have repeated values.

Now, we need to know whether x and y can represent the exact same value from the set.

If yes, then when 1 is in the set, so is 1+1 and so is thus 2+1.

However, if they cannot represent similar values, the set could just be {1}.

So we really need to read the question carefully. It states that “If both x and y are in S, then so is x + y”

Since the word “both’ is used, we already know that a set cannot have repeated values. 
hence, we will consider it a “yes”. If so, the statement is sufficient.
Correct Answer: D.

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