Difficulty: Easy
Correct Answer: Neither conclusion I nor conclusion II follows
Explanation:
Introduction / Context:
This syllogism question deals with two types of items, radios and table lamps, and a larger category, electric goods. You are told that both radios and table lamps belong to the class of electric goods, and you must determine whether any overlap between radios and table lamps is forced by the statements. This tests whether you understand that sharing a common superset does not automatically imply an overlap between the subsets.
Given Data / Assumptions:
Concept / Approach:
In set theory terms, the statements say Radios ⊂ ElectricGoods and TableLamps ⊂ ElectricGoods. This means both sets are contained within the larger electric goods set. However, the premises do not mention any direct relationship between radios and table lamps. We must decide whether this limited information forces an intersection between the two subsets. If we can draw a valid Venn diagram in which radios and table lamps do not overlap, then neither conclusion is logically necessary.
Step-by-Step Solution:
Step 1: Translate the statements into set language:
Step 2: Observe that the statements do not say that all electric goods are radios or table lamps. There is no statement such as “all electric goods are radios” or “all electric goods are table lamps.”
Step 3: Consider whether some radios must also be table lamps. You can easily imagine a store where radios and table lamps are entirely different products, both belonging to the broader category of electric goods. It is entirely possible that no individual item is simultaneously a radio and a table lamp.
Step 4: Construct a Venn diagram. Draw a large circle for ElectricGoods. Inside it, draw two smaller separate circles, one for Radios and one for TableLamps, ensuring they do not overlap. This diagram satisfies both statements, because every radio is inside ElectricGoods and every table lamp is inside ElectricGoods.
Step 5: In this diagram, Radios ∩ TableLamps is empty, meaning there is no object that is both a radio and a table lamp. This directly contradicts both Conclusion I and Conclusion II, which require at least some overlap between the two sets.
Verification / Alternative check:
The example above demonstrates that the given premises allow a situation with no overlap between radios and table lamps. For a conclusion like “some radios are table lamps” or “some table lamps are radios” to follow logically, it must hold in every model that satisfies the premises. Because we have found a valid model with zero overlap, these conclusions are not forced by the data, and hence neither conclusion follows.
Why Other Options Are Wrong:
Option a: “Only conclusion I follows” is incorrect because we have shown that there can be no overlap at all between the sets.
Option b: “Only conclusion II follows” is also incorrect for the same reason; Conclusion II is just a symmetric restatement of I.
Option d: “Both conclusion I and conclusion II follow” is clearly wrong; both depend on the existence of at least some overlapping items.
Option e: “Either conclusion I or conclusion II follows” is incorrect because neither is logically necessary.
Common Pitfalls:
Many learners mistakenly think that if two sets are both subsets of a third, they must overlap. This is not true in general. For example, dogs and cats are both subsets of animals, but an individual animal is usually not both a dog and a cat. Always check whether the premises explicitly assert an overlap; if they do not, and you can draw a disjoint diagram, then overlap based conclusions do not follow.
Final Answer:
Neither conclusion I nor conclusion II follows.
Discussion & Comments