Difficulty: Easy
Correct Answer: Neither conclusion I nor conclusion II follows
Explanation:
Introduction / Context:
This question involves three related objects from the hardware domain: bolts, nails, and screws. Although they are familiar physical items, you must ignore real world details and focus purely on the logical relationships described by the statements. The goal is to check whether the conclusions about all nails and all screws are justified by the given information.
Given Data / Assumptions:
- Statement I: All bolts are nails, so the set of bolts lies entirely inside the set of nails.
- Statement II: Some screws are bolts, meaning at least one element is both a screw and a bolt.
- Conclusion I: All nails are screws, which claims the entire nail set is inside the screw set.
- Conclusion II: All screws are nails, which claims the entire screw set is inside the nail set.
Concept / Approach:
Use set notation and Venn diagrams. Let B represent bolts, N represent nails, and S represent screws. From Statement I we get B ⊆ N. From Statement II we know S ∩ B is non empty. That means there are some screws that belong to N by virtue of being bolts. However, having some overlap is very different from saying that all elements of one set are inside another. We carefully test both universal conclusions against this structure.
Step-by-Step Solution:
Step 1: From B ⊆ N, every bolt is a nail. So if an object is in set B, it is automatically in set N.
Step 2: From S ∩ B being non empty, some screws are bolts. These particular screws are in both S and B.
Step 3: Since all bolts are nails, these screws that are bolts must also belong to N. So some screws are nails.
Step 4: However, nothing is said about screws that are not bolts. There could be many screws that are outside B and therefore might or might not be nails.
Step 5: Because we do not know whether every nail is a screw, the statement "All nails are screws" in Conclusion I is not forced by the information. Nails could include many items that are not screws.
Step 6: Similarly, we do not know whether every screw is inside N. We only know about some screws that are bolts, but other screws could be outside both B and N. Thus, we cannot conclude that "All screws are nails" as claimed in Conclusion II.
Verification / Alternative check:
Draw N as a large circle. Inside N draw a smaller circle B for bolts. Now draw S so that it intersects B in at least one region but extends outside N as well. This satisfies both statements because some screws are bolts and all bolts are nails. In this picture, clearly not all nails are screws and not all screws are nails, which shows that both conclusions are false in at least one valid configuration.
Why Other Options Are Wrong:
Option A assumes that all nails are screws, which is not supported. Option B assumes all screws are nails, which again is not supported. Option C claims both conclusions follow, which is stronger and therefore even further from the truth. Option E suggests that some screws are nails, which is true, but it does not match the exact requirement of the question, which focuses only on the given conclusions I and II. The correct answer must address those specific conclusions.
Common Pitfalls:
A common error is to overlook the word "some" in Statement II and mentally replace it with "all." This leads to the mistaken impression that all screws are bolts, and thus all screws are nails. Another trap is to read too much symmetry into the sets and assume that because one set is inside another, the reverse might also hold. Always treat "all," "some," and "no" very carefully, as they carry precise logical meaning.
Final Answer:
The correct option is Neither conclusion I nor conclusion II follows, because the statements only guarantee that some screws are nails through bolts, and nothing more general is logically forced.
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