Propositional and Predicate Logic - AI - Data Science & Artificial Intelligence Previous Year Questions

Let x and y be two propositions. Which of the following statements is a tautology /are tautologies?

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”?

Let \(p\) and \(q\) be any two propositions. Consider the following propositional statements.
\(S_1 : p \rightarrow q, \quad S_2 : \neg p \wedge q, \quad S_3 : \neg p \vee \bar{q}, \quad S_4 : \neg p \vee \neg q,\)
where \(\wedge\) denotes conjunction (AND operation), \(\vee\) denotes disjunction (OR operation), and \(\neg\) denotes negation (NOT operation). Which one of the following options is correct?
(Note: \(\equiv\) denotes logical equivalence)

Which of the following statements is NOT true?
(The names of the predicates are intuitive.)

Sentence X is said to entail Sentence Y if whenever X is TRUE, Y also must hold TRUE.
Which of the following statements is/are correct if X entails Y?

Assume that a Creative (\(C\)) person will Succeed (\(S\)) if the person is also Disciplined (\(D\)), but will not succeed otherwise. Now, consider the following statements:
(i) \(C \land S \Leftrightarrow D\)
(ii) \(C \Rightarrow (S \Leftrightarrow D)\)
(iii) \(C \Leftrightarrow ((D \Rightarrow S) \lor \neg S)\)
Which of the following options is correct?

Let \( P(x) \) be a predicate.
Which of the following statements is/are NOT valid in first-order logic?