Given below are two statements 1 and 2, and two conclusions I and II. Statement 1: All entrepreneurs are wealthy. Statement 2: All wealthy are risk seekers. Conclusion I: All risk seekers are wealthy. Conclusion II: Only some entrepreneurs are risk seekers. Based on the above statements and conclusions, which one of the following options is CORRECT?
Analyzing Syllogism Statements and Conclusions
We need to determine which conclusions logically follow from the given statements.
Statement Analysis
- Statement 1: All entrepreneurs are wealthy.
- Let E represent Entrepreneurs and W represent Wealthy people.
- In logical terms: $E \subseteq W$ (The set of Entrepreneurs is a subset of the set of Wealthy people).
- Statement 2: All wealthy are risk seekers.
- Let R represent Risk Seekers.
- In logical terms: $W \subseteq R$ (The set of Wealthy people is a subset of the set of Risk Seekers).
Deriving Conclusions
Combining Statement 1 and Statement 2 using the principle of transitivity:
- If $E \subseteq W$ and $W \subseteq R$, then it logically follows that $E \subseteq R$.
- This derived conclusion states: All entrepreneurs are risk seekers.
Evaluating Conclusion I
- Conclusion I: All risk seekers are wealthy.
- In logical terms: $R \subseteq W$.
- This is the converse of Statement 2 ($W \subseteq R$). The converse of a true universal affirmative statement ("All A are B") is not necessarily true.
- Example: All dogs (A) are mammals (B), but not all mammals (B) are dogs (A).
- Therefore, Conclusion I is incorrect.
Evaluating Conclusion II
- Conclusion II: Only some entrepreneurs are risk seekers.
- The phrase "Only some X are Y" typically implies two conditions: (1) Some X are Y, and (2) Some X are NOT Y.
- From our derived conclusion, we know that All entrepreneurs are risk seekers ($E \subseteq R$).
- This means the condition "Some entrepreneurs are NOT risk seekers" is false.
- Since one of the necessary conditions for "Only some entrepreneurs are risk seekers" is false, Conclusion II is incorrect.
Final Verdict
Both Conclusion I and Conclusion II are logically incorrect based on the given statements.
Therefore, the correct option is that neither conclusion I nor II is correct.
Important Questions from Syllogism
Six statements, S1 to S6, are given.
S1: All GATE candidates are engineers.
S2: Some GATE candidates are CAT candidates.
S3: All GRE candidates are GATE candidates.
S4: All engineers are GRE candidates.
S5: Some CAT candidates are GRE candidates.
S6: Some CAT candidates are engineers.
Based on S1-S6, following options provide a set of three statements, followed by a fourth statement. Identify the option(s) where the fourth statement can be necessarily inferred from the set of three statements.
Statements :
All pens are pencils.
No pencil is cap.
Conclusions:
(I) All caps are pencils.
(II) No pen is a cap.
Statements:
All huts are mansions.
All mansions are temples.
Conclusions :
(I) Some temples are huts.
(II) Some temples are mansions.
Statements:
1. All heroes are winners.
2. All winners are lucky people.
Inferences:
I. All lucky people are heroes.
II. Some lucky people are heroes.
III. Some winners are heroes.
Which of the above inferences can be logically deduced from statements 1 and 2?Human beings are one among many creatures that inhabit an imagined world. In this imagined world, some creatures are cruel. If in this imagined world, it is given that the statement “Some human beings are not cruel creatures” is FALSE, then which of the following set of statement(s) can be logically inferred with certainty?
(i) All human beings are cruel creatures.
(ii) Some human beings are cruel creatures.
(iii) Some creatures that are cruel are human beings.
(iv) No human beings are cruel creatures.
