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GATE DA 2024 Q54: First-order logic for 'all balls are round except rugby balls'

Question

Let game(ball, rugby) be true if the ball is used in rugby and false otherwise. Let shape(ball, round) be true if the ball is round and false otherwise.

Consider the following logical sentences:

s1: ∀\forall ball ¬\neggame(ball, rugby) ⇒\Rightarrow shape(ball, round)

s2: ∀\forall ball ¬\negshape(ball, round) ⇒\Rightarrow game(ball, rugby)

s3: ∀\forall ball game(ball, rugby) ⇒\Rightarrow ¬\negshape(ball, round)

s4: ∀\forall ball shape(ball, round) ⇒\Rightarrow ¬\neggame(ball, rugby)

Which of the following choices is/are logical representations of the assertion, "All balls are round except balls used in rugby"?

Try it: multiple select: one or more correct options

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Answer: A, C (official GATE 2024 final answer key)

Concept used: "Except" states both directions: non-rugby balls are round, and rugby balls are not round. An implication equals its contrapositive.

First step: s2 is the contrapositive of s1 and s4 is the contrapositive of s3, so the assertion is (s1 or s2) together with (s3 or s4).

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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