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.
Only III
Understanding Syllogism Statements and Conclusions
This question asks us to analyze given statements and determine which of the conclusions logically follow from them. We need to treat the statements as true, even if they seem different from real-world facts. This type of problem is based on the rules of logical deduction or syllogisms.
Analyzing the Given Statements
We have three statements:
- Statement I: All A are B. (Every member of set A is also a member of set B).
- Statement II: Some B are C. (There is at least one member of set B that is also a member of set C).
- Statement III: All C are D. (Every member of set C is also a member of set D).
Analyzing the Given Conclusions
We need to evaluate three conclusions:
- Conclusion I: Some A are D.
- Conclusion II: Some C are A.
- Conclusion III: Some B are D.
Evaluating Each Conclusion Based on Statements
Let's check each conclusion to see if it necessarily follows from the statements.
Evaluation of Conclusion I: Some A are D.
We need to find a link between A and D. The statements link A to B, B to C, and C to D.
- Statement I: All A are B.
- Statement II: Some B are C.
- Statement III: All C are D.
From Statement I and II ("All A are B" and "Some B are C"), we cannot derive any definite conclusion about the relationship between A and C. For example, it's possible that the 'Some B' that are 'C' do not overlap with the 'All A' that are 'B'.
Since we cannot establish a direct or indirect necessary link between A and C, we cannot establish a necessary link between A and D through C, even using Statement III ("All C are D"). Conclusion I, "Some A are D," is not necessarily true based on the statements. It might be possible, but it's not guaranteed.
Evaluation of Conclusion II: Some C are A.
This conclusion is related to Conclusion I, just in reverse. If "Some A are D" doesn't necessarily follow, then "Some C are A" is also unlikely to follow directly from the given statements. The link involves C to B and B to A.
- Statement III: All C are D. (Implies Some D are C)
- Statement II: Some B are C. (Implies Some C are B)
- Statement I: All A are B. (Implies Some B are A)
We have "Some C are B" and "All A are B". From "Some C are B" and "All A are B", we cannot logically conclude "Some C are A". The set C and the set A might both be subsets of B but have no overlap with each other. Conclusion II, "Some C are A," is not necessarily true.
Evaluation of Conclusion III: Some B are D.
We need to find a link between B and D. Statements II and III provide this link.
- Statement II: Some B are C.
- Statement III: All C are D.
Let's combine these two statements. "Some B are C" means there's an overlap between B and C. "All C are D" means that the entire set C is contained within the set D. If some part of B is in C, and all of C is in D, then that same part of B must also be in D.
This follows the pattern: "Some X are Y" and "All Y are Z" leads to the conclusion "Some X are Z".
Applying this pattern:
- X = B
- Y = C
- Z = D
So, "Some B are C" + "All C are D" logically leads to "Some B are D". Conclusion III necessarily follows from the statements.
Summary of Conclusions
- Conclusion I: Some A are D. - Does not follow.
- Conclusion II: Some C are A. - Does not follow.
- Conclusion III: Some B are D. - Follows.
Therefore, only Conclusion III logically follows from the given statements.
Important Questions from Statements and Conclusions
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.
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, 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.
