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?
This question requires analyzing logical deductions based on two given statements using syllogistic reasoning.
Statement Analysis
Let H represent 'Heroes', W represent 'Winners', and L represent 'Lucky People'.
- Statement 1: All heroes are winners. This can be written as $H \subseteq W$.
- Statement 2: All winners are lucky people. This can be written as $W \subseteq L$.
Combining these statements, we get a chain: $H \subseteq W \subseteq L$. This implies that all heroes are lucky people ($H \subseteq L$).
Inference Analysis
Inference I: All lucky people are heroes
This inference states $L \subseteq H$. Our derived relationship is $H \subseteq L$. The inference $L \subseteq H$ is the converse and does not logically follow from $H \subseteq L$. Therefore, Inference I is invalid.
Inference II: Some lucky people are heroes
This inference states that there is an overlap between 'Lucky People' and 'Heroes'. Since we deduced $H \subseteq L$ (All Heroes are Lucky People), it implies that if there exists at least one hero, then that hero is a lucky person. Thus, some lucky people (namely, the heroes) are heroes. Therefore, Inference II is valid.
Inference III: Some winners are heroes
This inference states that there is an overlap between 'Winners' and 'Heroes'. From Statement 1, we know $H \subseteq W$ (All Heroes are Winners). This implies that if there exists at least one hero, then that hero is a winner. Thus, some winners (namely, the heroes) are heroes. Therefore, Inference III is valid.
Conclusion
Based on the analysis, inferences II and III can be logically deduced from the given statements.
Valid Inferences: II and III
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.
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?
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.
