In the following questions, two statements S1 & S2 are given followed by two conclusions I & II. Taking the two statements S1 and S2 as true, decide which one of the answer (A), (B), (C), (D) logically follows.
Statements : S1 : Some players are brave. S2 : All boys are brave. Conclusions : I : Some players are boys. II : Some boys are players.
Neither I nor II follows.
To determine which conclusion logically follows from the given statements, we need to analyze each statement and conclusion carefully. We will use the principles of logical deduction, often visualized with Venn diagrams, to assess the validity of the conclusions. Understanding the relationship between the sets defined by the statements is key to solving these logical reasoning problems.
Statements Analysis
Let's first understand the two given statements. In these types of logical reasoning problems, it's helpful to define the sets involved:
| Set Symbol | Description |
|---|---|
| P | Players |
| B | Brave individuals |
| O | Boys |
- Statement S1: Some players are brave. This statement implies that there is at least one player who is also brave. It suggests an overlap or intersection between the set of 'players' and the set of 'brave' individuals. In terms of sets, this means the intersection of P and B is not empty: $P \cap B \neq \emptyset$.
- Statement S2: All boys are brave. This statement implies that the entire set of 'boys' is included within the set of 'brave' individuals. There are no boys who are not brave. In terms of sets, this means O is a subset of B: $O \subseteq B$.
Conclusions Evaluation
Now, let's evaluate each conclusion based on the combined information from S1 and S2. A conclusion logically follows only if it is necessarily true given the truth of the statements.
Conclusion I Evaluation: Some Players are Boys
- Conclusion I: Some players are boys. This means there is at least one player who is also a boy. In set notation, this implies the intersection of P and O is not empty: $P \cap O \neq \emptyset$.
- From Statement S1, we know that some players are brave ($P \cap B \neq \emptyset$).
- From Statement S2, we know that all boys are brave ($O \subseteq B$).
- Consider a scenario where both statements are true, but Conclusion I is false. Imagine the set of 'Brave' people as a large group. Inside this group, we have all 'Boys'. The set of 'Players' overlaps with the 'Brave' set. However, this overlap could happen entirely within the part of 'Brave' that is outside the 'Boys' set. For example, if 'Brave' people are those who like chocolate, 'Players' are those who play cricket, and 'Boys' are those who study in school.
- S1: Some cricket players (players) like chocolate (brave). This is possible.
- S2: All school students (boys) like chocolate (brave). This is also possible.
- Does it logically follow that some cricket players (players) are school students (boys)? Not necessarily. A cricket player who likes chocolate might not be a school student, and a school student who likes chocolate might not play cricket. There is no information in the statements that forces an overlap between the set of players and the set of boys.
- Therefore, Conclusion I does not logically follow.
Conclusion II Evaluation: Some Boys are Players
- Conclusion II: Some boys are players. This means there is at least one boy who is also a player. In set notation, this implies the intersection of O and P is not empty: $O \cap P \neq \emptyset$.
- This conclusion is essentially asking the same thing as Conclusion I, just stated in the reverse order (due to the commutative property of intersection, $A \cap B$ is the same as $B \cap A$). If 'some players are boys' is not necessarily true, then 'some boys are players' is also not necessarily true.
- The same counter-example and reasoning used for Conclusion I applies here. The statements do not provide sufficient information to establish a definite overlap between the set of 'boys' and the set of 'players'. The overlap of 'Players' with 'Brave' could occur in a segment of 'Brave' that has no intersection with 'Boys'.
- Therefore, Conclusion II does not logically follow.
Deduction Summary
Based on our detailed analysis, neither Conclusion I nor Conclusion II can be logically derived from the given statements. While it is possible for some players to be boys, or some boys to be players in a real-world scenario, the statements provided do not guarantee this relationship. In logical reasoning, a conclusion must necessarily follow from the premises for it to be considered valid.
Thus, the correct option is that neither Conclusion I nor Conclusion II follows.
Important Questions from Syllogism
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All dancers are talented.
Some girls are dancers.
Conclusions:
I. Some girls are talented.
II. All talented are girls.
III. All girls are talented.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All directors are actors.
No actor is a producer.
All choreographers are directors.
Conclusions:
I. No choreographer is producer.
II. Some actors are choreographers.
III. No director is a producer.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follows from the statements.
Statements:
All lemons are plums.
All plums are dates.
Some dates are mangoes.
Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
Some cards are postcards.
Some cards are envelopes.
All envelopes are copies.
Conclusions:
I. Some copies are envelopes.
II. Some postcards are copies.
III. Some cards are copies.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All employees are tax-payers.
Some employees are farmers.
Some farmers are doctors.
Conclusions:
I. No farmer is a tax-payer.
II. Some farmers are tax-payers.
