Three statements are given followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements. Statements: 1. All teachers are students. 2. All principal are teachers. 3. Some principals are doctors. Conclusions: I. Some students are teachers. II. No student is a doctor.
Understanding the Logic Puzzle: Statements and Conclusions
This question asks us to analyze a set of statements and determine which of the given conclusions logically follow from them, assuming the statements are true, even if they contradict common knowledge. This type of problem is known as a syllogism in logic.
Let's break down the statements provided:
- Statement 1: All teachers are students. (Every member of the set 'Teachers' is also a member of the set 'Students'. Mathematically, this can be represented as Teachers ⊂ Students).
- Statement 2: All principals are teachers. (Every member of the set 'Principals' is also a member of the set 'Teachers'. Mathematically, this can be represented as Principals ⊂ Teachers).
- Statement 3: Some principals are doctors. (There is at least one member who is both in the set 'Principals' and the set 'Doctors'. Mathematically, this can be represented as Principals ∩ Doctors ≠ ∅).
Deriving Inferences from the Statements
We can combine these statements to draw further conclusions:
- From Statement 2 ("All principals are teachers") and Statement 1 ("All teachers are students"), we can infer that All principals are students. (If A ⊂ B and B ⊂ C, then A ⊂ C. Here, A=Principals, B=Teachers, C=Students).
- From Statement 3 ("Some principals are doctors") and the inference "All principals are students" (derived above), we can infer that Some students are doctors. (If some A are B, and all A are C, then some C are B. Here, A=Principals, B=Doctors, C=Students).
- From Statement 2 ("All principals are teachers") and Statement 3 ("Some principals are doctors"), we can infer that Some teachers are doctors. (If some A are B, and all A are C, then some C are B. Here, A=Principals, B=Doctors, C=Teachers. Note: This inference is valid, but not directly needed to evaluate the given conclusions).
Evaluating the Conclusions Based on Statements and Inferences
Now let's evaluate each conclusion:
- Conclusion I: Some students are teachers.
Statement 1 says "All teachers are students". If every teacher is a student, it logically follows that there must be some students who are teachers (unless the set of teachers is empty, which is not implied). For example, if there is at least one teacher, that teacher is also a student, meaning 'some students' are 'teachers'. Thus, Conclusion I follows.
- Conclusion II: No student is a doctor.
From Statement 3, we know "Some principals are doctors". From our derived inference, we know "All principals are students". Combining these, if some principals are doctors, and all principals are students, then it must be true that some students are doctors. Conclusion II states the opposite ("No student is a doctor"). Since we've established that "Some students are doctors", Conclusion II does not follow.
- Conclusion III: Some students are principals.
From Statement 2 ("All principals are teachers") and Statement 1 ("All teachers are students"), we derived the inference "All principals are students". If every principal is a student, it logically follows that there must be some students who are principals (unless the set of principals is empty, which is not implied). If there is at least one principal, that principal is also a student, meaning 'some students' are 'principals'. Thus, Conclusion III follows.
Summary of Conclusion Validity
Based on our analysis:
- Conclusion I: Some students are teachers. (Follows)
- Conclusion II: No student is a doctor. (Does not follow)
- Conclusion III: Some students are principals. (Follows)
Therefore, only conclusions I and III logically follow from the given statements.
| Statement/Conclusion | Relationship | Validity |
|---|---|---|
| Statement 1 | All Teachers are Students | Given |
| Statement 2 | All Principals are Teachers | Given |
| Statement 3 | Some Principals are Doctors | Given |
| Inference 1 | All Principals are Students | Follows from 1 & 2 |
| Inference 2 | Some Students are Doctors | Follows from 3 & Inference 1 |
| Conclusion I | Some Students are Teachers | Follows from Statement 1 |
| Conclusion II | No Student is a Doctor | Contradicts Inference 2, Does Not Follow |
| Conclusion III | Some Students are Principals | Follows from Inference 1 |
Revision Table: Key Syllogism Rules
| Statement Type | Representation | Meaning |
|---|---|---|
| All A are B | A ⊂ B | Every A is a B. Implies Some A are B and Some B are A (if A is not empty). |
| No A are B | A ∩ B = ∅ | There is no overlap between A and B. Implies No B are A. |
| Some A are B | A ∩ B ≠ ∅ | There is at least one element common to A and B. Implies Some B are A. |
| Some A are not B | Subset of A is not in B | There is at least one element in A that is not in B. Does not imply Some B are not A. |
Additional Information: Syllogism Techniques
Syllogism problems can often be solved using Venn diagrams or simply by carefully analyzing the relationships between the categories mentioned in the statements.
- Venn Diagrams: Represent each category (Teachers, Students, Principals, Doctors) as overlapping circles. Shade regions based on the statements. For example, "All teachers are students" means the part of the 'Teachers' circle outside the 'Students' circle is empty. "Some principals are doctors" means there is an 'x' placed in the overlapping region of 'Principals' and 'Doctors'. Then, check if the conclusions are supported by the diagram.
- Statement Combination: Look for common terms between statements to link them. For example, the term 'teachers' links Statement 1 and Statement 2, allowing us to connect 'principals' and 'students'. The term 'principals' links Statement 2 and Statement 3, allowing us to connect 'teachers' and 'doctors', and also 'students' and 'doctors' via the derived 'principals are students' link.
- "All" implies "Some": A key rule is that if "All A are B", then "Some A are B" is true (assuming A exists). Also, if "All A are B", then "Some B are A" is true (assuming A exists and A is not empty). This is why Conclusion I ("Some students are teachers") follows from Statement 1 ("All teachers are students"). Similarly, Conclusion III ("Some students are principals") follows from the derived "All principals are students".
Similar Questions
- All flowers are plants. Some plants are flowering plants. Which conclusion is logically valid?
Two statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known f act s, decide which of the conclusions logically follow(s) from the statements.
Statements:
Some carpenters are singers.
All singers are doctors.
Conclusions:
I. No doctor is a carpenter.
II. Some doctors are singers.
III. Some carpenters are doctors.
Three statements are given, followed by three conclusions numbered I, II and III. Assuming the statement s to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
Some machines are coolers.
All coolers are robots.
Some microwaves are coolers.
Conclusions:
I. Some coolers are machines.
II. Some robots are machines.
III. No robot is a microwave.
Two statements are given followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
1. All apples are mangoes.
2. Some bananas are mangoes.
Conclusions:
I. Some mangoes are bananas.
II. Some mangoes are apples.
III. No mango is an apple.Three statements are given followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements.
Statements:
1. No bed is a chair.
2. All tables are beds.
3. No cupboard is a bed.
Conclusions:
I. No table is a chair.
II. No cupboard is a table.
III. Some beds are tables.
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:
1. No city is a district.
2. Some districts are zones.
3. Some zones are states.
Conclusions:
I. Some zones are districts.
II. Some districts are cities.
lII. No state is a zone.
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:
1. Some tables are chairs.
2. Some chairs are beds.
3. All beds are goods.
Conclusions:
I. Some chairs are goods.
Il. Some goods are not beds.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is ture, 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:
No man is engineer.
No engineer is driver.
Some drivers are shop-keepers.
Conclusions:
1. Some men are drivers.
2. No shop-keepers are engineer.
3. Some shop-keepers are not engineers.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is ture, 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 aero-planes are cars.
All helicopters are cars.
All cars are scooters.
Conclusions:
1. All helicopters are scooters.
2. Some aero-planes are helicopters.
3. No aero-plane is helicopter.
Read the given statements and conclusions carefully. Assuming that the information given in the statement 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 pillows are doors.
Some doors are beds.
Conclusions:
I. Some doors are pillows.
II. All beds are doors.
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:
Some students are players.
All players are male.
Conclusions:
I. All males are players.
II. Some males are students.
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:
No creative is an employer.
All experts are creative.
All workers are experts.
Conclusions:
I. No employer is an expert.
II. No worker is an employer.
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 actors are choreographers.
All choreographers are producers.
Not a single producer is a director.
Conclusions:
I. Some actors are directors.
II. Not a single actor is a director.
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 :
1. All flowers are tulips.
2. No tulips are whites.
Conclusions :
I. All tulips are flowers.
II. Some tulips are whites.
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 rats are dogs.
Some rats are hens.
Conclusions:
I. Some rats are dogs.
II. Some hens are rats.
