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: \(\forall \textit{ball}\ \neg\, \textit{game(ball, rugby)} \Longrightarrow \textit{shape(ball, round)}\)
s2: \(\forall \textit{ball}\ \neg\, \textit{shape(ball, round)} \Longrightarrow \textit{game(ball, rugby)}\)
s3: \(\forall \textit{ball}\ \textit{game(ball, rugby)} \Longrightarrow \neg\, \textit{shape(ball, round)}\)
s4: \(\forall \textit{ball}\ \textit{shape(ball, round)} \Longrightarrow \neg\, \textit{game(ball, rugby)}\)
Which of the following choices is/are logical representations of the assertion,
“All balls are round except balls used in rugby”?
Answer
(A), (C)
Explanation
The assertion says two things about every ball: a ball not used in rugby is round, and a ball used in rugby is not round.
- s1 states the first part, and s2 is its contrapositive (a ball that is not round is used in rugby), so s1 and s2 say the same thing.
- s3 states the second part, and s4 is its contrapositive, so s3 and s4 say the same thing.
A correct representation needs one sentence from each pair, so \(s1 \wedge s3\) (A) and \(s2 \wedge s3\) (C) both work. Option B repeats the first part and would still allow a round rugby ball; option D repeats the second part and would allow a non-rugby ball that is not round.
Answer: A, C