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Read the given statements and conclusions carefully. You have to take the given statements to be true even if they seem to be at variance from commonly known facts and decide which conclusion(s) logically follow(s) from the given statements. Statements: Some pens are sketch pens. All sketch pens are erasers. Some erasers are rulers. Conclusions: (I) Some rulers are sketch pens. (II) Some erasers are pens.
Only conclusion (I) follows.
Neither conclusion (I) nor (II) follows.
Both conclusions (I) and (II) follow.
Only conclusion (II) follows.
Only conclusion (II) follows.
Visualize Venn diagrams: Pens and Sketch Pens overlap; Sketch Pens are completely inside Erasers; Rulers overlap with Erasers. Conclusion I fails because Rulers only touch Erasers, not necessarily Sketch Pens. Conclusion II follows because if some pens are sketch pens and all sketch pens are erasers, then those specific pens are also erasers.
Visualize Venn diagrams: Pens and Sketch Pens overlap; Sketch Pens are completely inside Erasers; Rulers overlap with Erasers. Conclusion I fails because Rulers only touch Erasers, not necessarily Sketch Pens. Conclusion II follows because if some pens are sketch pens and all sketch pens are erasers, then those specific pens are also erasers.
Analyze Statement 1 & 2
Some pens are sketch pens (overlap). All sketch pens are erasers (sketch pens set is fully inside erasers set). Thus, those pens that are sketch pens are definitely erasers.
Analyze Statement 3 & Conclusion I
Some erasers are rulers. While rulers overlap with erasers, there is no direct link defined between rulers and sketch pens. Therefore, 'Some rulers are sketch pens' is a possibility, not a logical necessity.
Evaluate Conclusion II
Since all sketch pens are erasers and some pens are sketch pens, the overlap between pens and sketch pens resides inside the 'erasers' group. Therefore, it is certain that 'Some erasers are pens'.
A: Incorrect because both conclusions do not follow; I is invalid. B: Incorrect because conclusion (II) is logically true. C: Incorrect because conclusion (I) is not a definite consequence of the premises.
D is correct because conclusion (II) is logically derived from the transitive relationship between pens, sketch pens, and erasers, while conclusion (I) remains a possibility without definite proof.
In syllogisms, if a conclusion is only 'possible' but not 'necessary' based on the given statements, it is considered false. Always look for the most restrictive diagram.
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