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.
Only conclusion II follows.
Logical Deduction from Statements: Employees, Tax-payers, Farmers
Let's carefully analyze the given statements and conclusions. We are given three statements and two conclusions, and we need to determine which conclusion logically follows from the statements.
Analyzing the Statements
The statements provide information about the relationships between different groups of people:
- Statement 1: All employees are tax-payers. This means that the set of employees is entirely contained within the set of tax-payers. If someone is an employee, they must also be a tax-payer.
- Statement 2: Some employees are farmers. This indicates that there is an overlap between the set of employees and the set of farmers. There are individuals who belong to both groups.
- Statement 3: Some farmers are doctors. This shows there is an overlap between the set of farmers and the set of doctors. There are individuals who belong to both groups. Note that the relationship between doctors and employees or tax-payers is not directly given, nor is it necessarily implied by the statements.
We can use these statements to deduce relationships between the groups. From Statement 1 ("All employees are tax-payers") and Statement 2 ("Some employees are farmers"), we can infer a crucial relationship. Since some individuals are employees and farmers, and all employees are tax-payers, those specific individuals (who are both employees and farmers) must also be tax-payers.
Therefore, there are some individuals who are simultaneously employees, farmers, and tax-payers. This establishes that there is an overlap between the set of farmers and the set of tax-payers.
Analyzing the Conclusions
Now let's examine the given conclusions based on our understanding of the statements:
Conclusion I: No farmer is a tax-payer.
This conclusion claims that there is absolutely no overlap between the set of farmers and the set of tax-payers. However, our analysis of the statements revealed that there are individuals who are both employees and farmers, and since all employees are tax-payers, these individuals must also be tax-payers. This means there exist farmers who are also tax-payers.
Thus, Conclusion I directly contradicts the logical inference from the statements. Therefore, Conclusion I does not logically follow.
Conclusion II: Some farmers are tax-payers.
This conclusion states that there is at least one individual who is a member of both the set of farmers and the set of tax-payers. As established from the statements, there are individuals who are 'employee + farmer'. Since 'All employees are tax-payers', any individual who is an employee is also a tax-payer. Therefore, the individuals who are 'employee + farmer' are also 'tax-payer + farmer'.
This confirms that there are indeed some farmers who are tax-payers. This conclusion is consistent with and logically follows from the given statements.
Summary of Analysis
- Conclusion I: No farmer is a tax-payer. - Does not follow.
- Conclusion II: Some farmers are tax-payers. - Logically follows.
Based on the analysis, only Conclusion II logically follows from the given statements.
Revision Table: Logic Statements and Conclusions
| Statement/Conclusion | Content | Analysis | Follows? |
|---|---|---|---|
| Statement 1 | All employees are tax-payers. | Employees $\subseteq$ Tax-payers | N/A |
| Statement 2 | Some employees are farmers. | Employees $\cap$ Farmers $\neq \emptyset$ | N/A |
| Statement 3 | Some farmers are doctors. | Farmers $\cap$ Doctors $\neq \emptyset$ | N/A |
| Inference | From S1 & S2: Some (Employee + Farmer) are Tax-payers. | (Employees $\cap$ Farmers) $\subseteq$ Tax-payers $\implies$ Farmers $\cap$ Tax-payers $\neq \emptyset$ | N/A |
| Conclusion I | No farmer is a tax-payer. | States Farmers $\cap$ Tax-payers $= \emptyset$ | No (Contradicts Inference) |
| Conclusion II | Some farmers are tax-payers. | States Farmers $\cap$ Tax-payers $\neq \emptyset$ | Yes (Matches Inference) |
Additional Information on Syllogism and Logic
This type of problem belongs to the category of Syllogism or deductive reasoning based on categorical propositions. Categorical propositions relate two categories or sets.
- Universal Affirmative (A): All S are P (e.g., All employees are tax-payers).
- Universal Negative (E): No S are P (e.g., No farmer is a tax-payer).
- Particular Affirmative (I): Some S are P (e.g., Some employees are farmers).
- Particular Negative (O): Some S are not P.
In this problem, we used one Universal Affirmative (A) statement and two Particular Affirmative (I) statements to derive conclusions. A key principle used here is that if a group A is a subset of group B (All A are B), and there is some overlap between group C and group A (Some C are A), then there must also be some overlap between group C and group B (Some C are B).
Statement 1 (All Employees are Tax-payers) can be seen as Employees $\subseteq$ Tax-payers. Statement 2 (Some Employees are Farmers) means the intersection of Employees and Farmers is not empty. If an element is in (Employees $\cap$ Farmers), it is both an Employee and a Farmer. Since all Employees are Tax-payers, this element must also be a Tax-payer. Thus, the element is also in (Farmers $\cap$ Tax-payers). Therefore, the intersection of Farmers and Tax-payers cannot be empty, meaning "Some Farmers are Tax-payers" is true.
Understanding how these types of statements interact is fundamental to solving logic and syllogism problems.
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.
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.
