Statements:
All engineers are smart.
Some smart people are creative.
No creative person is dull.
Conclusions:
(1) Some smart people are not dull.
(2) Some engineers are creative.
(3) Some dull people are smart.
(4) No engineer is dull.
Understanding the Statements and Conclusions
This problem involves logical reasoning, specifically analyzing a set of statements and determining which conclusions can be logically derived from them. We need to assume the given statements are true, even if they seem contrary to real-world knowledge.
Analyzing the Statements
Let's represent the sets involved:
- E: Engineers
- S: Smart people
- C: Creative people
- D: Dull people
The statements can be translated into logical forms:
- Statement 1: All engineers are smart.
- This means the set E is entirely contained within the set S.
- In logical notation: \(\forall x (E(x) \implies S(x))\).
- Statement 2: Some smart people are creative.
- This means there is at least one person who is both smart and creative. The sets S and C overlap.
- In logical notation: \(\exists x (S(x) \land C(x))\).
- Statement 3: No creative person is dull.
- This means the set C and the set D are completely separate; they have no members in common.
- In logical notation: \(\forall x (C(x) \implies \neg D(x))\) or \(C \cap D = \emptyset\).
Analyzing the Conclusions
Now, let's evaluate each conclusion based on the statements:
Conclusion (1): Some smart people are not dull.
- From Statement 2, we know there exists at least one person (let's call them 'Alex') who is both Smart (S) and Creative (C). So, Alex \(\in S\) and Alex \(\in C\).
- From Statement 3, we know that anyone who is Creative (C) cannot be Dull (D). Since Alex is Creative, Alex cannot be Dull.
- Therefore, Alex is Smart (S) and Alex is not Dull (\(\neg D\)).
- This directly supports the conclusion that "Some smart people are not dull".
- Conclusion (1) logically follows.
Conclusion (2): Some engineers are creative.
- Statement 1 tells us all Engineers (E) are Smart (S).
- Statement 2 tells us some Smart people (S) are Creative (C).
- However, the smart people who are creative might be only those smart people who are *not* engineers. The statements do not guarantee that the overlap between S and C must include any members of E.
- For example, imagine the set S contains {Engineer1, Engineer2, NonEngineer1, NonEngineer2}. Let E = {Engineer1, Engineer2}. Let C = {NonEngineer1, NonEngineer2}. Here, All E are S, and Some S are C (specifically, NonEngineer1 and NonEngineer2), but no E are C.
- Conclusion (2) does not logically follow.
Conclusion (3): Some dull people are smart.
- Statement 3 states No Creative (C) is Dull (D).
- Statement 2 states Some Smart (S) are Creative (C).
- Statement 1 states All Engineers (E) are Smart (S).
- We know that the set D is separate from C. We also know S and C overlap. However, we have no information about the relationship between the set D (Dull) and the set S (Smart) or the set E (Engineers). It is possible that no dull people are smart, or all dull people are smart, or some are and some aren't. The given statements don't provide enough information to conclude that *some* dull people must be smart.
- Conclusion (3) does not logically follow.
Conclusion (4): No engineer is dull.
- We know All Engineers (E) are Smart (S).
- We know No Creative (C) is Dull (D).
- We know Some Smart people (S) are Creative (C).
- Consider an engineer. This engineer is smart. Can this engineer be dull?
- An engineer is smart. This smartness doesn't automatically make them creative. If an engineer is smart but *not* creative, then Statement 3 (No C is D) doesn't apply to them. Such a smart, non-creative engineer could potentially be dull.
- Since we can conceive of a scenario where an engineer is dull (e.g., a smart person who isn't creative could be dull), we cannot definitively conclude that *no* engineer is dull.
- Conclusion (4) does not logically follow.
Final Deduction
Based on the analysis of each conclusion:
- Conclusion (1) logically follows from the statements.
- Conclusions (2), (3), and (4) do not necessarily follow from the statements.
Therefore, only conclusion (1) is logically valid.
Similar Questions
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1. No bed is a chair.
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Conclusions:
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II. No cupboard is a table.
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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.
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1. Some tables are chairs.
2. Some chairs are beds.
3. All beds are goods.
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I. Some chairs are goods.
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No man is engineer.
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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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All experts are creative.
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Some actors are choreographers.
All choreographers are producers.
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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.
