GATE CE 2024 Set 2 — Question 2
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Question
Statements:
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?
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?
Correct answer
(B) Only II and III
Solution
Let be the set of heroes, be the set of winners, and be the set of lucky people.
From Statement 1: (All heroes are winners).
From Statement 2: (All winners are lucky people).
Combining these, we get .Evaluating Inferences:
I. All lucky people are heroes (): Not necessarily true. is the superset.
II. Some lucky people are heroes (): Since , if there is at least one hero, then some lucky people are definitely heroes. This is a valid logical deduction in standard syllogism.
III. Some winners are heroes (): Since , if there is at least one hero, then some winners are definitely heroes. This is also a valid logical deduction.Thus, only II and III follow.
From Statement 1: (All heroes are winners).
From Statement 2: (All winners are lucky people).
Combining these, we get .Evaluating Inferences:
I. All lucky people are heroes (): Not necessarily true. is the superset.
II. Some lucky people are heroes (): Since , if there is at least one hero, then some lucky people are definitely heroes. This is a valid logical deduction in standard syllogism.
III. Some winners are heroes (): Since , if there is at least one hero, then some winners are definitely heroes. This is also a valid logical deduction.Thus, only II and III follow.
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