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
In the question below are given some statements followed by some conclusions based on those statements. Take the given statements to be true even if they seem to be different from commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements.
Statements:
I. Some P are Q.
II. All P are N.
Conclusion:
I. All Q are N.
II. Some P are not Q.
In the question below are given some statements followed by some conclusions based on those statements. Take the given statements to be true even if they seem to be different from commonly known facts. Read all the conclusions and then decide which of the given conclusions logically follows from the given statements.
Statements:
I. Some P are T.
II. Some T are S.
III. No J is P.
Conclusion:
I. Some S are not J.
II. Some T are not J.
III. Some J are not T.
In the question below are given some statements followed by some conclusions based on those statements. Take 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 conclusions logically follows from the given statements.
Statement :
I. All A are B.
II. No B is D.
III. Some D are G.
Conclusion :
I. No A is D.
II. Some G are not B.
III. Some A are not G.
In the question below are given some statements followed by some conclusions based on those statements. Take 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 conclusions logically follows from the given statements.
Statements:
I. All A are B.
II. Some B are C.
III. All C are D.
Conclusion:
I. Some A are D.
II. Some C are A.
III. Some B are D.
In the question below, some statements followed by some conclusions based on those statements are given. Take 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 conclusions logically follows from the given statements.
Statements:
I. All P are Q.
II. Some P are C.
Conclusions:
I. Some Q are C.
II. Some P are not C.
