Read the given statements and conclusions carefully. Assuming that the information given in the statements in 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: 1. Some dancers are writers. 2. All writers are students. Conclusions: I. Some students are dancers. II. Some students are writers.
Only conclusions I and II follows
Solving Syllogism Statements and Conclusions
This question asks us to analyze given statements and determine which of the provided conclusions logically follow. This type of problem is known as syllogism in logical reasoning. We need to assume the statements are true, even if they contradict common knowledge, and then deduce the validity of the conclusions based solely on these statements.
Analyzing the Statements
We have two statements:
- Some dancers are writers.
- All writers are students.
Let's represent these relationships. Statement 1 tells us there is an overlap between the group of 'dancers' and the group of 'writers'. Statement 2 tells us that the entire group of 'writers' is contained within the group of 'students'.
Visualizing with Venn Diagrams
Venn diagrams are a helpful tool to visualize the relationships between sets described in syllogism problems.
- Draw a circle for 'Dancers'.
- Draw a circle for 'Writers' that partially overlaps with the 'Dancers' circle (representing "Some dancers are writers").
- Draw a larger circle for 'Students' that completely encloses the 'Writers' circle (representing "All writers are students").
When you draw this, you'll see that the portion where 'Dancers' and 'Writers' overlap is also contained within the 'Students' circle because the entire 'Writers' circle is inside the 'Students' circle.
Evaluating the Conclusions
Now, let's examine each conclusion based on our understanding of the statements and the Venn diagram:
Conclusion I: Some students are dancers.
We know from Statement 1 that some dancers are writers. We also know from Statement 2 that all writers are students. Since the dancers who are writers are also students (because all writers are students), it means there are some individuals who are both dancers and students. Therefore, some students are dancers logically follows from the statements.
Conclusion II: Some students are writers.
Statement 2 explicitly says "All writers are students". If all writers are students, it necessarily implies that there are some students who are writers (unless the set of writers is empty, which isn't implied). Therefore, some students are writers logically follows from the statements.
Conclusion III: No students is a dancer.
From our analysis of Conclusion I, we found that some students *are* dancers. This conclusion directly contradicts that finding. If some students are dancers, then it is impossible for no student to be a dancer. Therefore, this conclusion does not follow from the statements.
Summary of Conclusions
Based on our analysis:
- Conclusion I: Some students are dancers - Follows
- Conclusion II: Some students are writers - Follows
- Conclusion III: No students is a dancer - Does Not Follow
Thus, only conclusions I and II logically follow from the given statements.
| Conclusion | Logically Follows? | Reasoning |
|---|---|---|
| I. Some students are dancers. | Yes | Dancers who are writers must also be students. |
| II. Some students are writers. | Yes | If all writers are students, some students must be writers. |
| III. No students is a dancer. | No | Contradicts Conclusion I, which follows. |
Matching with Options
We found that only conclusions I and II follow. Let's look at the options provided:
- Option 1: Either conclusion I or III follows
- Option 2: Only conclusions II and III follows
- Option 3: Only conclusions I and II follows
- Option 4: All conclusions I, II and III follows
Option 3 matches our finding that only conclusions I and II follow.
Revision Table: Key Syllogism Rules
| Statement Type | Example | Interpretation |
|---|---|---|
| All A are B | All pens are blue. | Set A is completely inside Set B. Implies Some B are A (if A is not empty). |
| No A is B | No cat is dog. | Set A and Set B are completely separate. Implies No B is A. |
| Some A are B | Some books are red. | There is at least one element common to Set A and Set B. Implies Some B are A. |
| Some A are not B | Some fruits are not sweet. | There is at least one element in Set A that is not in Set B. Does not imply anything about Some B are not A. |
Additional Information: Understanding Syllogisms
Syllogism is a form of logical reasoning where a conclusion is drawn from two or more premises (statements). The statements provide information about the relationships between different categories or groups. The conclusion is then evaluated to see if it is necessarily true based *only* on the information given in the statements.
Key points to remember when solving syllogism problems:
- Assume Statements are True: Treat the given statements as absolute facts, regardless of real-world accuracy.
- Look for Necessary Conclusions: A conclusion follows only if it *must* be true based on the statements. If there is any possibility, however remote, that the conclusion is false, it does not follow.
- Use Visual Aids: Venn diagrams are highly recommended for visualizing the relationships between the categories mentioned in the statements.
- Be Careful with "Some": "Some" means "at least one". It can range from "just one" to "all". In standard syllogism, "Some A are B" implies "at least one A is B".
Practicing with different combinations of statements and conclusions helps build proficiency in solving these logical reasoning questions.
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.
