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
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.
The statements below are followed by two conclusions labelled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow (s) from the information given in the statements.
Statements:
All books are grapes.
Some grapes are fruits.
Conclusions:
I. No grape is a book.
II. Some grapes are books.The statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
Some fans are radio.
All fans are light.
Conclusion I: Some radios are light.
Conclusion II: Some lights are radio.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 scissors are spoons.
All spoons are plates.
Conclusions:
1. Some plates are spoons.
2. Some spoons are scissors.
3. Some plates are scissors.
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:
No boat is ship.
Some ships are rods.
Conclusions:
I. No boat is rod.
II. Some rods are ships.
III. Some rods are not boats.
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 bottles are glasses.
All glasses are jugs.
Conclusions:
I. All bottles are jugs.
II. All jugs are bottles.
III. Some glasses are bottles.
In the following question below are given some statements followed by some conclusions. 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 pen are black.
II. Some pen are blue.
III. No black is red.
Conclusions:
I. No pen is red.
II. Some blue are black.
III. Some black are pens.In the following question below are given some statements followed by some conclusions. 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. All cups are plates.
II. All plates are red.
Conclusions:
I. All cups are red.
II. All plates are cups.
III. All red are plates.In the following question below are given some statements followed by some conclusions. 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 sky are blue.
II. Some blue are soft.
Conclusions:
I. Some sky are soft.
II. Some soft are sky.What will come in the place of the question mark (?) in the following equation if '+' and '-' are interchanged and 'x' and '÷' are interchanged?
108 × 2 − 14 ÷ 6 + 27 = ?
Important Questions from Syllogism
The statements below are followed by conclusions labeled I, II and III. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
All twenty are thirty.
All thirty are forty.
All forty are sixty.
All sixty are seventy.
Conclusions:
I. Some forty are thirty.
II. Some seventy are sixty.
III. No thirty is twenty.The statements below are followed by two conclusions labelled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
All Strong are animals.
Some animals are Tigers.
All Tigers are Sharp.
Conclusions:
I. Some Strong are Sharp.
II. No Strong is Sharp.The statements below are followed by conclusions labelled I, II and III. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.Statements:
Some women are weak.
Some weaks are female.
All female are iron.
All iron are gold.Conclusions:
I. Some weaks are iron.
II. Some gold are weaks.
III. Some women are female.
The statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.Statements:
Some teaspoons are glasses.
All teddies are teaspoons.
Conclusions:
I. Some teddies are glasses.
II. Some glasses are teddies.
The statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally Established facts, decide which conclusion(s) logically and definitely follow(s) from the Information, Given in the Statements.
Statements:
All bangles are rings.
Some rings are toys.
Some toys are dolls.
Conclusions:
I. Some rings are bangles.
II. Some dolls are rings.
