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.
None of the conclusions follows
Solving Statements and Conclusions Logic Questions
This question asks us to analyze given statements and determine which of the provided conclusions logically follow from them. We must assume the statements are true, even if they contradict general knowledge.
Analyzing the Statements
We are given two statements:
- Statement I: Some P are Q.
- Statement II: All P are N.
Let's understand what these statements mean using a diagrammatic approach, like Venn diagrams.
- 'Some P are Q' implies there is an overlap between the set of P and the set of Q. There is at least one element that belongs to both P and Q. It does not mean that ONLY some P are Q; it is possible that all P are Q or all Q are P, or even all P and all Q are the same set, while still satisfying 'Some P are Q'.
- 'All P are N' implies that the entire set of P is contained within the set of N. Every element of P is also an element of N.
Analyzing the Conclusions
Now let's evaluate each conclusion based on the statements:
Conclusion I: All Q are N.
From Statement I, we know 'Some P are Q'. From Statement II, we know 'All P are N'. Since some part of Q is P, that part of Q must also be in N because all P are in N. However, Statement I only guarantees that *some* Q are P. It doesn't give any information about the part of Q that is *not* P. This part of Q might or might not be in N. Therefore, we cannot definitively conclude that 'All Q are N'.
Consider an example: Let P be {apples, bananas}, Q be {apples, cherries}, and N be {apples, bananas, oranges}.
- Statement I: Some P are Q (apples are in both). True.
- Statement II: All P are N (apples and bananas are in N). True.
- Conclusion I: All Q are N (apples and cherries are in N). False, because cherries are in Q but not necessarily in N based on the statements.
Conclusion I does not logically follow from the given statements.
Conclusion II: Some P are not Q.
Statement I says 'Some P are Q'. As discussed earlier, the phrase 'Some P are Q' in logic means "There exists at least one P that is Q". It does not preclude the possibility that "All P are Q".
For example, if P = {1, 2, 3} and Q = {1, 2, 3}, then 'Some P are Q' is true (e.g., 1 is P and Q). In this case, 'All P are Q' is also true. If 'All P are Q' is true, then 'Some P are not Q' is false.
Since 'Some P are Q' is consistent with 'All P are Q', and if 'All P are Q' is true, 'Some P are not Q' is false, we cannot definitively conclude that 'Some P are not Q' must be true based only on Statement I. Statement II ('All P are N') does not provide any information that would force 'Some P are not Q' to be true either.
Conclusion II does not logically follow from the given statements.
Conclusion
Based on the analysis of both conclusions, neither Conclusion I nor Conclusion II 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 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.
