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. All A are B. II. Some B are C. Conclusions: I. Some A are C. II. Some A are not C. III. Some A are B.
Only conclusion III follows
This question requires analyzing logical statements and determining which conclusions can be drawn from them. We are given two statements and three potential conclusions. We need to treat the statements as true, even if they contradict common knowledge.
Analyzing the Statements
Let's break down the given statements:
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Statement I: All A are B.
This is a universal affirmative statement. It means that every member of category A is also a member of category B. In set notation, this can be represented as $A \subseteq B$. This implies that the set A is entirely contained within the set B.
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Statement II: Some B are C.
This is a particular affirmative statement. It means there is at least one member that belongs to both category B and category C. There is an overlap between sets B and C. In set notation, this is $\exists x (B(x) \land C(x))$.
Evaluating the Conclusions
Now, let's evaluate each conclusion based on the statements:
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Conclusion I: Some A are C.
Statement I tells us that all A are within B ($A \subseteq B$). Statement II tells us that some B overlap with C. However, the overlap between B and C might occur in the part of B that does not contain A. We cannot be certain that any A are part of this overlap. Therefore, Conclusion I does not logically follow from the given statements.
Using Venn diagrams, if circle A is inside circle B, and circle C overlaps with circle B, the overlap between C and B might be entirely outside of A.
In logical notation: $A \subseteq B$ and $\exists x (B(x) \land C(x))$ does not necessarily imply $\exists x (A(x) \land C(x))$.
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Conclusion II: Some A are not C.
Similar to Conclusion I, we don't have enough information to conclude this. It's possible that all A fall into the subset of B that also overlaps with C. We cannot definitively say that some A are outside of C. Therefore, Conclusion II does not logically follow.
In logical notation: $A \subseteq B$ and $\exists x (B(x) \land C(x))$ does not necessarily imply $\exists x (A(x) \land \neg C(x))$.
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Conclusion III: Some A are B.
Statement I is "All A are B". This is a universal statement. A universal affirmative statement ("All X are Y") logically implies the corresponding particular affirmative statement ("Some X are Y"), provided that the subject category (X) is not empty. In these types of reasoning problems, we assume that the categories mentioned (like A) are not empty. Therefore, if "All A are B" is true, it must also be true that "Some A are B".
In logical notation: $A \subseteq B$ implies $\exists x (A(x) \land B(x))$, assuming set A is non-empty.
Conclusion III logically follows directly from Statement I.
Final Determination
Based on the analysis:
- Conclusion I: Does not follow.
- Conclusion II: Does not follow.
- Conclusion III: Follows.
Therefore, only Conclusion III logically follows from the given statements.
Important Questions from Statements and Conclusions
Directions: The questions has a statement followed by two conclusions I and II. You have to decide which of the conclusions follows from the given statement and mark your answer as
Statement: Mr A is not among the candidates shortlisted for the post of vice chancellor of a university
Conclusions :
I) Mr. A will be elected as Vice - chancellor
II) Mr. A will not be considered for the post of Vice-chancellor
Read the given statements and conclusions carefully. Assuming that the information in the statements is true, even if it appears to be at variance from commonly established facts, decide which of the conclusions from / logically follows these statements.
Statement:
Poverty makes man hardworking.
Hardworking people are not poor.
Conclusion:
I. A person must be poor to be hardworking.
II. Diligence helps to remove poverty.
Read the given statements and conclusions carefully and decide which of the given conclusions logically follows from the given statements.
Statement:
A asked C, "What time is it now?" Is it half past two? ''
Conclusions:
1. A wants to go to lunch.
2. A wants to confirm what the time is because his watch is off.
In the question one statement is given, followed by two conclusions, I and II. You have to consider the statements to be true even if it seems to be at variance from commonly known facts. You have to decide which of the given conclusions, if any, follows from the given statements.
Statement : 55% of Manisha's office colleagues went to watch the movie.
Conclusion I : Manisha went to watch the movie.
Conclusion II : Manisha did not go to watch the movie.
In the question two statements are given, followed by two conclusions, I and II. You have to consider the statements to be true even if it seems to be at variance from commonly known facts. You have to decide which of the given conclusions, if any, follows from the given statements.
Statement 1 : All schools are books.
Statement 2 : No books are papers.
Conclusion I : No schools are books.
Conclusion II : Some books are schools.
