The question consists of a problem question followed by two statements I and…

A data-sufficiency statement is sufficient only when it forces one single, definite answer to the question, using only the rules already given — with no ambiguity left over.

Set S obeys two closure rules: (a) if p is in S, then 1/p is in S; (b) if p and q are in S, then p + q is in S. Rule (b) applies to any two members already known to be in S — including the same member used twice (p = q), since a single known element trivially satisfies "both p and q are in S." So once one value is in S, its double is forced into S as well.

Checking Statement I: 1/5 is in S

1. Apply rule (a) with p = 1/5: its reciprocal, 1/(1/5) = 5, must be in S.

2. So Statement I alone forces "5 is in S" — a single, definite answer.

Checking Statement II: 1/2 is in S

1. Apply rule (b) with p = q = 1/2: 1/2 + 1/2 = 1 must be in S.

2. Apply rule (a) with p = 1/2: its reciprocal, 2, must be in S.

3. Apply rule (b) with p = q = 2: 2 + 2 = 4 must be in S.

4. Apply rule (b) with p = 4 and q = 1 (both already forced into S above): 4 + 1 = 5 must be in S.

5. So Statement II alone also forces "5 is in S" — a single, definite answer.

Cross-check

Both chains are forced, not merely possible: starting from 1/5, rule (a) leaves no alternative to 5 being in S; starting from 1/2, the sequence 1/2 → 1 (via rule (b) on itself) → 2 (via rule (a)) → 4 (via rule (b) on itself) → 5 (via rule (b) with 4 and 1) is likewise forced at every step, with no closure rule left unused that could produce a different or ambiguous outcome.

Since Statement I alone forces a definite yes, and Statement II alone also forces a definite yes (through a longer chain that uses rule (b) on a value together with itself), each statement independently is sufficient on its own. So either of the statements is sufficient in answering the problem question.