GATE 2024 DA – Question 54
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: $\forall ball\ \neg game(ball, rugby) \Rightarrow shape(ball, round)$
s2: $\forall ball\ \neg shape(ball, round) \Rightarrow game(ball, rugby)$
s3: $\forall ball\ game(ball, rugby) \Rightarrow \neg shape(ball, round)$
s4: $\forall ball\ shape(ball, round) \Rightarrow \neg game(ball, rugby)$
Which of the following choices is/are logical representations of the assertion, "All balls are round except balls used in rugby"?
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Show answer and explanation
Correct answer: (A) $s1 \wedge s3$; (C) $s2 \wedge s3$
Explanation
The assertion says that a ball that is not used in rugby is round (s1), and that a ball used in rugby is not round (s3). Now s2 is the contrapositive of s1, so it says the same thing, and s4 is the contrapositive of s3. So $s1 \wedge s3$ and $s2 \wedge s3$ both express the assertion fully. $s1 \wedge s2$ says only the first part, and $s3 \wedge s4$ says only the second part.