Statements: All keys are locks. All locks are bangles. All bangles are cars. Conclusions: a. Some cars are locks. b. Some bangles are keys. c. Some cars are keys. Which of the above conclusions logically follow from the given statements?

Introduction / Context:
This is a classic syllogism question involving three categorical statements about sets: keys, locks, bangles and cars. You must determine which of the listed conclusions necessarily follow. Such questions test your understanding of subset relations and how information flows through chains like “All A are B, all B are C”.

Given Data / Assumptions:

All keys are locks (Keys ⊂ Locks).
All locks are bangles (Locks ⊂ Bangles).
All bangles are cars (Bangles ⊂ Cars).
Conclusions: a. Some cars are locks.
b. Some bangles are keys.
c. Some cars are keys.

Concept / Approach:
Whenever we have “All A are B”, we treat A as a subset of B. Combining multiple subset relations allows us to see indirect connections. Also, when we say “Some X are Y”, it simply means there is at least one element common to both sets, provided the subset is non-empty. In examination logic, we assume that each mentioned set (keys, locks, bangles, cars) has at least one member unless explicitly stated otherwise.

Step-by-Step Solution:
From “All keys are locks” and “All locks are bangles”, we get Keys ⊂ Locks ⊂ Bangles. From “All bangles are cars”, we further extend the chain: Keys ⊂ Locks ⊂ Bangles ⊂ Cars. Conclusion a: “Some cars are locks.” Since the set of locks is a subset of cars, and locks are assumed to exist, at least some cars are indeed locks. So a follows. Conclusion b: “Some bangles are keys.” Since keys are a subset of locks, and locks are a subset of bangles, all keys are also bangles. Therefore, there are some bangles that are keys. So b follows. Conclusion c: “Some cars are keys.” From the chain, keys are also a subset of cars (via locks and bangles). Thus, some cars are keys. So c follows.

Verification / Alternative check:
You can picture a nested diagram: keys inside locks, locks inside bangles, bangles inside cars. Every key is inside all three larger sets. Therefore, there must automatically be cars that are locks, bangles that are keys, and cars that are keys. All three conclusions naturally follow from the nesting structure.

Why Other Options Are Wrong:
Any option that omits one of the conclusions (like “only a and b follow” or “only c follows”) fails to recognise that each conclusion is a direct consequence of the subset chain. “None of these” is incorrect because we have clearly identified that all three statements do follow logically.

Common Pitfalls:
Students sometimes think that “some” requires separate information. But if one set is fully contained inside another, and we assume that set is non-empty, then “some of the larger set are elements of the smaller set” is always true. Confusing the direction of subset relations (for example, thinking “all bangles are locks”) is another common mistake.

Final Answer:
All three conclusions a, b and c follow logically. The correct option is a, b and c follow.