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.
Only conclusions I and II follow
Understanding Syllogism Statements and Conclusions
This question asks us to analyze three given statements and determine which of the three conclusions logically follow from them, assuming the statements are true.
The Given Statements:
- Some machines are coolers.
- All coolers are robots.
- Some microwaves are coolers.
The Given Conclusions:
- Some coolers are machines.
- Some robots are machines.
- No robot is a microwave.
Analyzing the Statements and Conclusions using Logic and Venn Diagrams
We can analyze each statement and conclusion using basic logic rules or by visualizing them with Venn diagrams.
Statement 1: Some machines are coolers.
This means there is an overlap between the set of Machines and the set of Coolers. If "Some A are B", then it logically follows that "Some B are A".
Statement 2: All coolers are robots.
This means the entire set of Coolers is contained within the set of Robots.
Statement 3: Some microwaves are coolers.
This means there is an overlap between the set of Microwaves and the set of Coolers.
Let's evaluate each conclusion based on these statements:
Conclusion I: Some coolers are machines.
This conclusion is the converse of Statement 1 ("Some machines are coolers"). If some machines belong to the category of coolers, then it must also be true that some coolers belong to the category of machines. This conclusion logically follows from Statement 1.
Conclusion II: Some robots are machines.
We know from Statement 1 that "Some machines are coolers". We also know from Statement 2 that "All coolers are robots".
If we have a group of machines that are also coolers (from Statement 1), and every single cooler is a robot (from Statement 2), then the group of machines that are also coolers must necessarily be robots as well.
Therefore, "Some machines are robots" follows. The converse of "Some machines are robots" is "Some robots are machines", which is Conclusion II. So, Conclusion II logically follows from Statements 1 and 2 combined.
Conclusion III: No robot is a microwave.
We know from Statement 3 that "Some microwaves are coolers". We also know from Statement 2 that "All coolers are robots".
If we have a group of microwaves that are also coolers (from Statement 3), and every single cooler is a robot (from Statement 2), then the group of microwaves that are also coolers must necessarily be robots as well.
Therefore, "Some microwaves are robots" logically follows from Statements 2 and 3 combined.
Conclusion III states "No robot is a microwave". This contradicts the logically derived conclusion "Some microwaves are robots". Therefore, Conclusion III does not follow.
Summary of Conclusions:
- Conclusion I: Some coolers are machines. (Follows)
- Conclusion II: Some robots are machines. (Follows)
- Conclusion III: No robot is a microwave. (Does not follow)
Based on our analysis, only conclusions I and II logically follow from the given statements.
Revision Table: Syllogism Analysis
| Statement/Conclusion | Relationship | Logical Follows? | Reasoning |
|---|---|---|---|
| Statement 1: Some machines are coolers. | Machines <-> Coolers (Some) | N/A | Given statement |
| Statement 2: All coolers are robots. | Coolers -> Robots (All) | N/A | Given statement |
| Statement 3: Some microwaves are coolers. | Microwaves <-> Coolers (Some) | N/A | Given statement |
| Conclusion I: Some coolers are machines. | Coolers <-> Machines (Some) | Yes | Converse of Statement 1 |
| Conclusion II: Some robots are machines. | Robots <-> Machines (Some) | Yes | From Statements 1 & 2 (Some A are B, All B are C implies Some A are C, thus Some C are A) |
| Conclusion III: No robot is a microwave. | Robots -/- Microwaves (None) | No | Contradicts result from Statements 2 & 3 (All B are C, Some A are B implies Some A are C, thus Some C are A). Some robots are microwaves. |
Additional Information: Syllogism Rules
Syllogisms are a form of deductive reasoning. Understanding basic rules and how to combine statement types is key to solving them. Here are a few points:
- Converse: "Some A are B" has a valid converse "Some B are A". "All A are B" does *not* have a valid converse "All B are A" or "Some B are A" without additional information. "No A is B" has a valid converse "No B is A".
- Combining Statements: When combining two statements, a valid conclusion can sometimes be drawn, especially when they share a middle term (like 'coolers' in this problem). The quality (affirmative/negative) and quantity (all/some/no) of the conclusion depend on the statements.
- Venn Diagrams: Drawing Venn diagrams for each statement and combining them can visually show overlaps and containments, making it easier to check conclusions.
Practicing with different types of statements and conclusions helps build proficiency in solving syllogism problems.
Similar Questions
- All flowers are plants. Some plants are flowering plants. Which conclusion is logically valid?
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Some carpenters are singers.
All singers are doctors.
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I. No doctor is a carpenter.
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III. Some carpenters are doctors.
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2. All principal are teachers.
3. Some principals are doctors.
Conclusions:
I. Some students are teachers.
II. No student is a doctor.
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Statements:
1. All apples are mangoes.
2. Some bananas are mangoes.
Conclusions:
I. Some mangoes are bananas.
II. Some mangoes are apples.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
