Given below are two statements and four conclusions drawn based on the statements. Statement 1: Some bottles are cups. Statement 2: All cups are knives. Conclusion I: Some bottles are knives. Conclusion II: Some knives are cups. Conclusion III: All cups are bottles. Conclusion IV: All knives are cups. Which one of the following options can be logically inferred? A. Only conclusion I and conclusion II are correct. B. Only conclusion II and conclusion III are correct. C. Only conclusion II and conclusion IV are correct. D. Only conclusion III and conclusion IV are correct.

GATE 2022 · General Aptitude · Statements Follow · medium

Answer: Conclusions I and II are correct; III and IV are not. The answer is A.

1. Chain statements to test Conclusion I: Some bottles are cups (Stmt 1) + All cups are knives (Stmt 2). Let X be an object in Bottles∩Cups. Since X is a cup and all cups are knives, X is also a knife. So X is in Bottles∩Knives => Some bottles are knives. Conclusion I is CORRECT.
2. Test Conclusion II: Some knives are cups: From Stmt 2: All cups are knives. Cups set is non-empty (Stmt 1 says some bottles are cups, so cups exist). Therefore there exist objects that are simultaneously cups and knives => Some knives are cups. Conclusion II is CORRECT.
3. Test Conclusion III: All cups are bottles: Stmt 1 says 'Some bottles are cups', not 'All cups are bottles'. The converse is not licensed. Conclusion III is INCORRECT.
4. Test Conclusion IV: All knives are cups: Stmt 2 says 'All cups are knives' (Cups ⊆ Knives). Reversing this to 'All knives are cups' (Knives ⊆ Cups) is the converse fallacy. Conclusion IV is INCORRECT.