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Syllogism-Based Data Sufficiency

Introduction

Syllogism-based Data Sufficiency problems यह पूछते हैं कि दिए गए statement(s) किसी दिये गये conclusion को logically स्थापित करने के लिए पर्याप्त हैं या नहीं। यहाँ conclusion को prove करना उद्देश्य नहीं है - आपको यह तय करना होता है कि कोई statement अकेले या दोनों मिलकर उस conclusion को निश्चित रूप से साबित करते हैं या नहीं।

यह pattern महत्वपूर्ण है क्योंकि यह precise logical inference सिखाता है: क्या वास्तव में “must be true” है और क्या केवल “may be true” है।

Pattern: Syllogism-Based Data Sufficiency

Pattern

Verbal statements को categorical relations (All, No, Some, Some not) में बदलें। फिर formal syllogistic rules से conclusion test करें: यदि कोई statement अकेला conclusion को guarantee करता है, तो वह sufficient है; अगर केवल दोनों मिलकर guarantee करें, तो दोनों आवश्यक; अगर कोई भी नहीं कर पाए, तो insufficient।

Useful translations:
All A are B → A ⊂ B
No A are B → A ∩ B = ∅
Some A are B → A ∩ B ≠ ∅ (existence implied)
Some A are not B → not(A ⊂ B) + existence implied

Step-by-Step Example

Question

क्या conclusion "All A are C" follow करता है?
(I) All A are B.
(II) All B are C.

Options:
A. केवल (I) पर्याप्त है
B. केवल (II) पर्याप्त है
C. प्रत्येक statement अकेला पर्याप्त है
D. दोनों statements साथ में आवश्यक हैं

Solution

  1. Step 1: (I) का विश्लेषण

    (I) ⇒ All A are B → A ⊂ B. B और C का कोई संबंध नहीं दिया, इसलिए A ⊂ C नहीं कहा जा सकता → (I) insufficient.

  2. Step 2: (II) का विश्लेषण

    (II) ⇒ All B are C → B ⊂ C. A और B का कोई संबंध नहीं दिया - A ⊂ C नहीं कहा जा सकता → (II) insufficient.

  3. Step 3: Combine

    (I) A ⊂ B और (II) B ⊂ C → transitivity से A ⊂ C ⇒ All A are C. दोनों together sufficient हैं।

  4. Final Answer:

    दोनों statements साथ में आवश्यक हैं → Option D

  5. Quick Check:

    A→B और B→C मिलकर ही A→C मिलता है; अकेले किसी से नहीं ✅

Quick Variations

1. Universal + particular mix: जैसे All A are B और Some C are A - ध्यान दें existence implied है।

2. Negative premises: No A are B + All B are C → implications set relations से सावधानी से निकालें।

3. Existential pitfalls: "Some" में existence अनिवार्य है; केवल universal premises से existence साबित नहीं होती।

4. Chain reasoning: कई universals (All) को जोड़कर universal conclusion तक पहुँचना।

Trick to Always Use

  • Step 1: हर statement को set-relations (All/No/Some) में translate करें।
  • Step 2: Transitivity जांचें: All A→B और All B→C ⇒ All A→C।
  • Step 3: “Some” वाले conclusions के लिए existence जरूरी है; premises में existence है या नहीं, यह देखें।
  • Step 4: यदि कोई भी statement अकेला set-relation को guarantee कर दे, वही sufficient है; अन्यथा combination चेक करें।

Summary

Summary

  • Premises को standard categorical relations में बदलकर ही conclusion test करें।
  • Universal premises पर transitivity लागू करें: A→B और B→C ⇒ A→C।
  • "Some" conclusions हमेशा existence require करते हैं - universal केवल अस्तित्व साबित नहीं कर सकते।
  • कभी शक हो तो counterexample बनाकर देखें - premises true हों लेकिन conclusion false हो सके तो वह insufficient है।

याद रखने वाला उदाहरण:
(I) All A are B और (II) All B are C ⇒ दोनों मिलकर All A are C; कोई भी अकेला sufficient नहीं।

Practice

(1/5)
1. Does the conclusion 'All A are B' follow?<br>(I) All A are B.<br>(II) Some B are C.
easy
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) explicitly states All A are B ⇒ A ⊂ B, so the conclusion 'All A are B' follows directly from (I). → Sufficient.

  2. Step 2: Analyze (II)

    (II) Some B are C gives information about B and C only and does not tell us anything about A → Insufficient.

  3. Final Answer:

    Only (I) is sufficient → Option A

  4. Quick Check:

    (I) gives the required inclusion explicitly; (II) is unrelated to A → ✅
Hint: If a statement repeats the conclusion verbatim (All X are Y), it is immediately sufficient.
Common Mistakes: Looking for transitive links when the conclusion is already stated.
2. Does the conclusion 'All B are C' follow?<br>(I) Some A are B.<br>(II) All B are C.
easy
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) Some A are B only states an overlap between A and B; it gives no information about B and C → Insufficient.

  2. Step 2: Analyze (II)

    (II) directly states All B are C ⇒ B ⊂ C, so the conclusion follows from (II) alone → Sufficient.

  3. Final Answer:

    Only (II) is sufficient → Option B

  4. Quick Check:

    (II) is the exact universal inclusion required for the conclusion → ✅
Hint: A direct universal premise matching the conclusion is sufficient by itself.
Common Mistakes: Treating partial overlap (Some) as if it implies universal inclusion.
3. Does the conclusion 'All C are D' follow?<br>(I) All C are D.<br>(II) All C are D.
easy
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) states All C are D ⇒ C ⊂ D, so the conclusion follows from (I) alone → Sufficient.

  2. Step 2: Analyze (II)

    (II) also states All C are D ⇒ (II) alone is likewise sufficient.

  3. Final Answer:

    Each statement alone is sufficient → Option C

  4. Quick Check:

    Both premises give the same universal inclusion required by the conclusion → ✅
Hint: If each premise independently provides the same universal relation, both are individually sufficient.
Common Mistakes: Looking for combination when single-premise universals already suffice.
4. Does the conclusion 'Some A are C' follow?<br>(I) Some A are B.<br>(II) All B are C.
medium
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) Some A are B indicates overlap A-B but gives no information about C → Insufficient.

  2. Step 2: Analyze (II)

    (II) All B are C ⇒ B ⊂ C but gives no information about A → Insufficient.

  3. Step 3: Combine

    From (I) Some A are B and from (II) All B are C ⇒ those A that are B are also C ⇒ Some A are C → Sufficient together.

  4. Final Answer:

    Both statements together are necessary → Option D

  5. Quick Check:

    Overlap (Some) + inclusion (All) ⇒ overlap propagates to C → ✅
Hint: A 'Some' overlap combined with an 'All' inclusion gives the required 'Some' conclusion.
Common Mistakes: Assuming 'All' alone implies existence for 'Some' conclusions without an overlap premise.
5. Does the conclusion 'Some B are A' follow?<br>(I) All B are A.<br>(II) There exists at least one B.
medium
A. Only (I) is sufficient
B. Only (II) is sufficient
C. Each statement alone is sufficient
D. Both statements together are necessary

Solution

  1. Step 1: Analyze (I)

    (I) All B are A ⇒ B ⊂ A. This is a universal inclusion but does not guarantee that any B actually exists (universal statements do not imply existence). Therefore (I) alone is insufficient to conclude 'Some B are A'.

  2. Step 2: Analyze (II)

    (II) There exists at least one B ⇒ ensures existence of B but gives no information about A. Thus (II) alone is insufficient to conclude 'Some B are A'.

  3. Step 3: Combine

    Combining (I) and (II): (I) provides B ⊂ A and (II) provides that B is non-empty. Together these imply that at least one member of B exists and, since B ⊂ A, that member is in A. Hence 'Some B are A' follows from the pair of statements.

  4. Final Answer:

    Both statements together are necessary → Option D

  5. Quick Check:

    Universal inclusion + explicit existence ⇒ particular conclusion (Some) ✅
Hint: A universal inclusion (All X are Y) needs an existence assertion to conclude a 'Some' statement.
Common Mistakes: Assuming 'All B are A' implies there exists B; universal claims do not imply existence.

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