The question asks us to identify which set of three given statements logically implies the fourth statement.

Analysis of Option 2: {S4, S1, S2} => S5

We need to determine if statements S4, S1, and S2 necessarily lead to statement S5.

• S4: All engineers are GRE candidates. This can be written as: $Engineers \subseteq GRE$.
• S1: All GATE candidates are engineers. This can be written as: $GATE \subseteq Engineers$.
• S2: Some GATE candidates are CAT candidates. This means the intersection is not empty: $GATE \cap CAT \neq \emptyset$.
• S5: Some CAT candidates are GRE candidates. This means the intersection is not empty: $CAT \cap GRE \neq \emptyset$.

Using the principle of transitivity:

• From $Engineers \subseteq GRE$ (S4) and $GATE \subseteq Engineers$ (S1), we can conclude that $GATE \subseteq GRE$.
• This means all GATE candidates are also GRE candidates.

Statement S2 says that there is at least one candidate who is both a GATE candidate and a CAT candidate. Let's call this candidate 'X'.

• So, X is a GATE candidate ( $X \in GATE$ ).
• And, X is a CAT candidate ( $X \in CAT$ ).

From step 2, we know that if someone is a GATE candidate, they are also a GRE candidate ($GATE \subseteq GRE$).

• Since X is a GATE candidate, X must also be a GRE candidate ( $X \in GRE$ ).
• We already know X is a CAT candidate ( $X \in CAT$ ).
• Therefore, X is both a CAT candidate and a GRE candidate.
• This confirms that the intersection of CAT and GRE candidates is not empty ($CAT \cap GRE \neq \emptyset$), which is exactly statement S5.

Conclusion: The set of statements {S4, S1, S2} necessarily implies S5.

Verification of Other Options (Brief)

A similar analysis can be performed for the other options. For instance:

• Option 1: {S1, S2, S3} => S5. S3 ($GRE \subseteq GATE$) combined with S1 ($GATE \subseteq Engineers$) gives $GRE \subseteq Engineers$. S2 ($GATE \cap CAT \neq \emptyset$) does not guarantee an overlap between CAT and GRE.
• Option 3: {S6, S1, S3} => S5. S1 ($GATE \subseteq Engineers$) and S3 ($GRE \subseteq GATE$) give $GRE \subseteq Engineers$. S6 ($CAT \cap Engineers \neq \emptyset$) does not guarantee an overlap between CAT and GRE.
• Option 4: {S4, S3, S1} => S6. S4 ($Engineers \subseteq GRE$), S3 ($GRE \subseteq GATE$), S1 ($GATE \subseteq Engineers$) implies $Engineers = GRE = GATE$. However, no statement connects CAT candidates to this group, so S6 ($CAT \cap Engineers \neq \emptyset$) cannot be inferred.

Thus, only Option 2 provides a valid inference.