Statement 1: All students are inquisitive.
Statement 2: Some students are inquisitive.
Statement 3: No student is inquisitive.
Statement 4: Some students are not inquisitive.
From the given four statements, find the two statements that CANNOT BE TRUE simultaneously, assuming that there is at least one student in the class.
Identifying Statements That Cannot Be True Simultaneously
The question requires identifying which pair of statements from the given four cannot be true at the same time, given there is at least one student.
Let 'S' represent the set of students and 'I' represent the property of being inquisitive.
- Statement 1: All students are inquisitive (All S are I).
- Statement 2: Some students are inquisitive (Some S are I).
- Statement 3: No student is inquisitive (No S are I).
- Statement 4: Some students are not inquisitive (Some S are not I).
Analyzing Statement Pairs
We need to examine the logical relationship between the statements in each option:
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Option 1: Statement 1 and Statement 3
- Statement 1 (All S are I) asserts that the set of students is fully contained within the set of inquisitive individuals.
- Statement 3 (No S are I) asserts that the set of students and the set of inquisitive individuals are completely disjoint (have no common members).
- These two statements are contraries. They directly oppose each other. If all students are inquisitive, it's impossible for no students to be inquisitive. Conversely, if no students are inquisitive, it's impossible for all of them to be inquisitive.
- Therefore, Statement 1 and Statement 3 cannot be true simultaneously.
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Option 2: Statement 1 and Statement 2
- Statement 1 (All S are I) implies Statement 2 (Some S are I), assuming there is at least one student. If all students possess the quality, then it logically follows that at least one student possesses it.
- These statements can be true simultaneously (in fact, if Statement 1 is true, Statement 2 must be true).
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Option 3: Statement 2 and Statement 4
- Statement 2 (Some S are I) means at least one student is inquisitive.
- Statement 4 (Some S are not I) means at least one student is not inquisitive.
- These statements are subcontraries. They cannot both be false simultaneously (because if one is false, the other must be true, given at least one student exists), but they can both be true simultaneously (if the class has both inquisitive and non-inquisitive students).
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Option 4: Statement 3 and Statement 4
- Statement 3 (No S are I) implies Statement 4 (Some S are not I), assuming there is at least one student. If no students are inquisitive, then it logically follows that at least one student is not inquisitive.
- These statements can be true simultaneously (if Statement 3 is true, Statement 4 must also be true).
Conclusion
Based on the analysis, Statement 1 and Statement 3 are contrary statements and thus cannot be true simultaneously.
Important Questions from Syllogism
- Consider the following sentences:
All benches are beds .No bed is a bulb. Some bulbs are lamps.
Which of the following can be inferred?
i. Some beds are lamps.
ii. Some lamps are beds. Given below are two statements followed by two conclusions. Assuming these statements to be true, decide which one logically follows.
Statements:
I. No manager is a leader.
II. All leaders are executives.
Conclusions:
I. No manager is an executive.
II. No executive is a manager.Given below are two statements 1 and 2, and two conclusions I and II.
Statement 1: All entrepreneurs are wealthy.
Statement 2: All wealthy are risk seekers.
Conclusion I: All risk seekers are wealthy.
Conclusion II: Only some entrepreneurs are risk seekers.
Based on the above statements and conclusions, which one of the following options is CORRECT?
Statements:
1. All heroes are winners.
2. All winners are lucky people.
Inferences:
I. All lucky people are heroes.
II. Some lucky people are heroes.
III. Some winners are heroes.
Which of the above inferences can be logically deduced from statements 1 and 2?Human beings are one among many creatures that inhabit an imagined world. In this imagined world, some creatures are cruel. If in this imagined world, it is given that the statement “Some human beings are not cruel creatures” is FALSE, then which of the following set of statement(s) can be logically inferred with certainty?
(i) All human beings are cruel creatures.
(ii) Some human beings are cruel creatures.
(iii) Some creatures that are cruel are human beings.
(iv) No human beings are cruel creatures.
