Six statements, S1 to S6, are given. S1: All GATE candidates are engineers. S2: Some GATE candidates are CAT candidates. S3: All GRE candidates are GATE candidates. S4: All engineers are GRE candidates. S5: Some CAT candidates are GRE candidates. S6: Some CAT candidates are engineers. Based on S1-S6, following options provide a set of three statements, followed by a fourth statement. Identify the option(s) where the fourth statement can be necessarily inferred from the set of three statements.
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.
1. Represent the statements logically:- 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.
Important Questions from Syllogism
Statements:
1. All heroes are winners.
2. All winners are lucky people.
Inferences:
I. All lucky people are heroes.
II. Some lucky people are heroes.
III. Some winners are heroes.
Which of the above inferences can be logically deduced from statements 1 and 2?Human beings are one among many creatures that inhabit an imagined world. In this imagined world, some creatures are cruel. If in this imagined world, it is given that the statement “Some human beings are not cruel creatures” is FALSE, then which of the following set of statement(s) can be logically inferred with certainty?
(i) All human beings are cruel creatures.
(ii) Some human beings are cruel creatures.
(iii) Some creatures that are cruel are human beings.
(iv) No human beings are cruel creatures.- Given below are four statements.
Statement 1: All students are inquisitive.
Statement 2: Some students are inquisitive.
Statement 3: No student is inquisitive.
Statement 4: Some students are not inquisitive.
From the given four statements, find the two statements that CANNOT BE TRUE simultaneously, assuming that there is at least one student in the class. - Based only on the truth of the statement ‘Some humans are intelligent', which one of the following options can be logically inferred with certainty?
Given below are two statements and four conclusions drawn based on the statements.
Statement 1: Some bottles are cups.
Statement 2: All cups are knives.
Conclusion I: Some bottles are knives.
Conclusion II: Some knives are cups.
Conclusion III: All cups are bottles.
Conclusion IV: All knives are cups.
Which one of the following options can be logically inferred?
