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.
Only conclusion I follows
Understanding Statements and Conclusions in Logic
This question asks us to analyze given statements and determine which of the provided conclusions logically follow from them. This is a common type of question in logical reasoning, often solved using methods like Venn diagrams or rules of syllogism.
Analyzing the Given Statements
We are given two statements:
- Statement 1: All dancers are talented.
- Statement 2: Some girls are dancers.
We must assume these statements are true, even if they contradict general knowledge.
Analyzing the Given Conclusions
We need to evaluate three conclusions based on the statements:
- Conclusion I: Some girls are talented.
- Conclusion II: All talented are girls.
- Conclusion III: All girls are talented.
Step-by-Step Evaluation of Conclusions
Let's examine each conclusion to see if it necessarily follows from the statements.
Evaluation of Conclusion I: Some girls are talented.
Statement 1 tells us that the set of "dancers" is completely inside the set of "talented" people. Statement 2 tells us that there is an overlap between the set of "girls" and the set of "dancers".
If some girls are dancers (Statement 2), and all dancers are talented (Statement 1), then those specific girls who are dancers must also be talented. Therefore, there are some girls who are talented.
This can be visualized with Venn diagrams:
| Set | Relation | Implication |
|---|---|---|
| Dancers (D) | Subset of Talented (T) | Every element in D is also in T |
| Girls (G) | Overlap with Dancers (D) | There is at least one element in G that is also in D |
| Result | Overlap between Girls (G) and Talented (T) | If an element is in the G and D overlap, and all D is in T, that element must also be in T. Since the G and D overlap is non-empty, the G and T overlap is non-empty. |
Conclusion I logically follows from the given statements.
Evaluation of Conclusion II: All talented are girls.
Statement 1 says all dancers are talented. It does not say that only girls or only dancers are talented. There could be other talented people (e.g., singers, artists) who are not dancers and not girls.
The statements only guarantee that the set of dancers is within the set of talented people and that some girls are within the set of dancers. They do not provide enough information to conclude that the entire set of talented people is contained within the set of girls.
Conclusion II does not logically follow from the given statements.
Evaluation of Conclusion III: All girls are talented.
Statement 2 says "Some girls are dancers". This implies that not all girls are necessarily dancers. Since only dancers are guaranteed by the statements to be talented (from Statement 1), we cannot conclude that girls who are not dancers are also talented.
The statements only tell us about the girls who are dancers. They say nothing about the talent of girls who are not dancers.
Conclusion III does not logically follow from the given statements.
Summary of Conclusions
Based on our analysis:
- Conclusion I (Some girls are talented) follows.
- Conclusion II (All talented are girls) does not follow.
- Conclusion III (All girls are talented) does not follow.
Therefore, only conclusion I logically follows from the given statements.
Revision Table: Statements and Conclusions
| Statements | Conclusions | Logically Follows? | Reasoning |
|---|---|---|---|
| 1. All dancers are talented. 2. Some girls are dancers. |
I. Some girls are talented. | Yes | Girls who are dancers are also talented. Since some girls are dancers, some girls must be talented. |
| 1. All dancers are talented. 2. Some girls are dancers. |
II. All talented are girls. | No | Talented people could exist who are not dancers or not girls. Statements don't exclude this. |
| 1. All dancers are talented. 2. Some girls are dancers. |
III. All girls are talented. | No | Only girls who are dancers are confirmed talented by the statements. Statements don't say anything about the talent of girls who are not dancers. |
Additional Information: Syllogism and Validity
This question is an example of a categorical syllogism problem. A syllogism is a type of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. In this case, we have two premises (the statements) and we are testing the validity of possible conclusions.
A conclusion logically follows from statements if and only if it is impossible for the statements to be true and the conclusion false simultaneously. In other words, the conclusion must be a necessary consequence of the statements.
The structure of the valid conclusion (Conclusion I) matches a common pattern in syllogisms:
All P are Q
Some R are P
Therefore, Some R are Q
Where P = Dancers, Q = Talented, R = Girls.
Understanding these basic structures and how sets relate to each other (inclusion, overlap, separation) is key to solving statements and conclusions problems accurately.
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 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.
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. Some L are R.
II. Some A are R.
Conclusion:
I. All A are L.
II. All R are L.
