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GATE 2024 IN GA Q.2: solution and answer

GATE 2024 · INGA Q.2 · 1 mark · MCQ

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?

  1. (A)

    Only I and II

  2. (B)

    Only II and III

  3. (C)

    Only I and III

  4. (D)

    Only III

See the question in the paper
GATE 2024 IN GA question 2Tap to open full size

Answer and solution

ANSWER

Only inferences II and III follow, so the answer is (B) Only II and III.

Official answer: BSource: Official answer key, GATE 2024 IN (IISc Bengaluru)

  1. Combine the two statements. "All heroes are winners" and "All winners are lucky people" are both universal affirmatives, and they chain to give: All heroes are lucky people. So .

  2. Draw it as a Venn diagram: the Heroes circle sits inside the Winners circle, which sits inside the Lucky-people circle. As usual in GATE syllogism questions, each class named in a statement is taken to have at least one member, so the Heroes circle is not empty.

  3. Inference I, "All lucky people are heroes", is the converse of a universal statement. A universal affirmative cannot be converted simply: the Lucky-people circle may contain people outside the Heroes circle. So I does not follow.

  4. Inference III, "Some winners are heroes": "All heroes are winners" converts by limitation (A to I) into "Some winners are heroes". The Heroes circle is non-empty and lies inside the Winners circle, so at least one winner is a hero. III follows.

  5. Inference II, "Some lucky people are heroes": the derived conclusion "All heroes are lucky people" also converts by limitation, giving "Some lucky people are heroes". Equivalently, the heroes who are winners (from III) are all lucky people, by statement 2. II follows.

  6. Only II and III follow, which is option (B).

KEY IDEA

Chain the two universal statements into Heroes ⊆ Winners ⊆ Lucky people. An "All A are B" statement can never be reversed to "All B are A". It can be converted by limitation to "Some B are A", since GATE treats every named class as non-empty.

WATCH OUT FOR

Students go wrong in two ways. Some reverse "All heroes are lucky people" into "All lucky people are heroes" and accept I, which leads to (A) or (C). Others argue that "All" never implies "Some" because the Heroes set might be empty, and choose (D) or reject II. GATE follows the traditional convention that universal statements are about classes with members, so both II and III follow.

From the GATE 2024 IN paper. Every question of this paper · Download the full paper · All 41 Reasoning questions, with notes

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