Some tables are shelves. Some shelves are chairs. All chairs are benches. Which of the following
conclusions can be deduced from the preceding sentences?
i.
At least one bench is a table
11.
At least one shelf is a bench
111.
At least one chair is a table
iv.
All benches are chairs
Options:
(A) Only i
(B) Only ii
(C) Only 11 and 111
(D) Only iv
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2017 Q.60)**
**1. Governing Concept & Question Classification**:
- **Section**: General Aptitude (GA)
- **Question Type**: **MCQ** (1 Mark)
- **Topic**: Logical Reasoning & Syllogism (Deductive Logic)
**2. Detailed Step-by-Step Analysis**:
- We are given three premises:
1. *Some tables are shelves.* ($T \cap S \neq \emptyset$)
2. *Some shelves are chairs.* ($S \cap C \neq \emptyset$)
3. *All chairs are benches.* ($C \subseteq B$)
- Let us evaluate each proposed conclusion:
- **Conclusion i: "At least one bench is a table"**:
- Shelves that are tables may be completely disjoint from shelves that are chairs. Thus, $B \cap T$ can be empty. This conclusion does not necessarily follow.
- **Conclusion ii: "At least one shelf is a bench"**:
- By Premise 2, there exist elements in $S$ that are also in $C$ (i.e., some shelves are chairs).
- By Premise 3, every element in $C$ is in $B$ ($C \subseteq B$).
- Therefore, the shelves that are chairs are guaranteed to be benches:
$S \cap C \neq \emptyset \text{ and } C \subseteq B \implies S \cap B \neq \emptyset$
- Thus, at least one shelf is definitely a bench. This conclusion **definitely follows**.
- **Conclusion iii: "At least one chair is a table"**:
- No direct relationship connects chairs and tables. Does not follow.
- **Conclusion iv: "All benches are chairs"**:
- Premise 3 states $C \subseteq B$ (All chairs are benches), which does not imply the converse $B \subseteq C$. Does not follow.
- Therefore, **Only conclusion ii** can be deduced.
- **Option-by-Option Evaluation**:
- **Option (A) [INCORRECT]**: Conclusion i cannot be deduced.
- **Option (B) [CORRECT]**: Only conclusion ii logically follows from the premises.
- **Option (C) [INCORRECT]**: Includes invalid conclusion iii.
- **Option (D) [INCORRECT]**: Conclusion iv commits the fallacy of affirming the converse.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **B**
- **Scoring Criteria**: +1.0 Mark (Negative marking: -0.33 marks).
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