GATE 2024 DA Q.54: solution and answer

GATE 2024 · DAQ.54 · 2 marks · MSQ

Let $g am e (ba l l, r ug b y)$ be true if the ball is used in rugby and false otherwise. Let $s ha p e (ba l l, r o u n d)$ be true if the ball is round and false otherwise. Consider the following logical sentences:

s1: $\forall ba l l \neg g am e (ba l l, r ug b y) \Rightarrow s ha p e (ba l l, r o u n d)$
s2: $\forall ba l l \neg s ha p e (ba l l, r o u n d) \Rightarrow g am e (ba l l, r ug b y)$
s3: $\forall ba l l g am e (ba l l, r ug b y) \Rightarrow \neg s ha p e (ba l l, r o u n d)$
s4: $\forall ba l l s ha p e (ba l l, r o u n d) \Rightarrow \neg g am e (ba l l, r ug b y)$

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

(A)
$s 1 \land s 3$
(B)
$s 1 \land s 2$
(C)
$s 2 \land s 3$
(D)
$s 3 \land s 4$

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Answer and solution

ANSWER

The correct choices are (A) and (C).

Official answer: A, CSource: Official answer key, GATE 2024 DA (IISc Bengaluru)

Let $G$ = game(ball, rugby) and $R$ = shape(ball, round). The assertion "All balls are round except balls used in rugby" says two things: a ball that is not used in rugby is round, and a ball used in rugby is not round.
Formalise the two parts: $\forall ba l l (\neg G \to R)$ and $\forall ba l l (G \to \neg R)$ . These are exactly $s 1$ and $s 3$ .
Contrapositive: $\neg G \to R \equiv \neg R \to G$ , so $s 2$ is logically equivalent to $s 1$ .
Contrapositive: $G \to \neg R \equiv R \to \neg G$ , so $s 4$ is logically equivalent to $s 3$ .
Hence $s 1 \land s 3$ is the full assertion, so (A) is correct. Since $s 2 \equiv s 1$ , we get $s 2 \land s 3 \equiv s 1 \land s 3$ , so (C) is also correct.
$s 1 \land s 2 \equiv s 1$ only says non-rugby balls are round. It never says rugby balls are not round, so (B) is incomplete. $s 3 \land s 4 \equiv s 3$ only says rugby balls are not round. It says nothing about non-rugby balls, so (D) is incomplete.

KEY IDEA

\"All X are Y except Z\" means two implications: not-Z implies Y, and Z implies not-Y. Each implication is equivalent to its contrapositive, so $s 1 \equiv s 2$ and $s 3 \equiv s 4$ . A pair is correct exactly when it contains one sentence from ${s 1, s 2}$ and one from ${s 3, s 4}$ .

WATCH OUT FOR

Treating $s 1 \land s 2$ or $s 3 \land s 4$ as the full assertion. Each of these pairs holds two equivalent sentences, so it states only one half of the assertion. Another slip is to miss that $s 2$ is the contrapositive of $s 1$ and so drop option (C).

From the GATE 2024 DA paper. Every question of this paper · Download the full paper · All 7 Logic questions, with notes

GATE 2024 DA Q.54: solution and answer

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