Question:
Let \( p \) and \( q \) be any two propositions. Consider the following propositional statements.
S1: \( p \rightarrow q \), S2: \( \neg p \land q \), S3: \( \neg p \lor q \), S4: \( \neg p \lor \neg q \)
where \( \land \) denotes conjunction (AND operation), \( \lor \) denotes disjunction (OR operation), and \( \neg \) denotes negation (NOT operation).
(Note: \( \equiv \) denotes logical equivalence)
Which one of the following options is correct?
Let \( p \) and \( q \) be any two propositions. Consider the following propositional statements.
S1: \( p \rightarrow q \), S2: \( \neg p \land q \), S3: \( \neg p \lor q \), S4: \( \neg p \lor \neg q \)
where \( \land \) denotes conjunction (AND operation), \( \lor \) denotes disjunction (OR operation), and \( \neg \) denotes negation (NOT operation).
(Note: \( \equiv \) denotes logical equivalence)
Which one of the following options is correct?
S1: \( p \rightarrow q \), S2: \( \neg p \land q \), S3: \( \neg p \lor q \), S4: \( \neg p \lor \neg q \)
where \( \land \) denotes conjunction (AND operation), \( \lor \) denotes disjunction (OR operation), and \( \neg \) denotes negation (NOT operation).
(Note: \( \equiv \) denotes logical equivalence)
Which one of the following options is correct?
Show Hint
Implications \( p \rightarrow q \) are logically equivalent to disjunctions \( \neg p \lor q \). Use truth tables to verify logical equivalences between logical operations.
Updated On: Feb 11, 2026
- \( S_1 \equiv S_3 \)
- \( S_2 \equiv S_3 \)
- \( S_2 \equiv S_4 \)
- \( S_1 \equiv S_4 \)
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Analyze the statements.
S1: \( p \rightarrow q \) is logically equivalent to \( \neg p \lor q \) by the definition of implication.
S2: \( \neg p \land q \) represents a conjunction, and is not logically equivalent to the other expressions.
S3: \( \neg p \lor q \) is logically equivalent to \( p \rightarrow q \), which matches S1.
S4: \( \neg p \lor \neg q \) is different and is not logically equivalent to S1 or S3.
Thus, S1 is logically equivalent to S3, making Option (A) the correct answer.
S1: \( p \rightarrow q \) is logically equivalent to \( \neg p \lor q \) by the definition of implication.
S2: \( \neg p \land q \) represents a conjunction, and is not logically equivalent to the other expressions.
S3: \( \neg p \lor q \) is logically equivalent to \( p \rightarrow q \), which matches S1.
S4: \( \neg p \lor \neg q \) is different and is not logically equivalent to S1 or S3.
Thus, S1 is logically equivalent to S3, making Option (A) the correct answer.
Was this answer helpful?
1
1
Top Questions on Logic gates
Which logic gate is represented by the following combinations of logic gates?
- WBJEE - 2026
- Physics
- Logic gates
- For the given logic gate, find the output function.
- JEE Main - 2026
- Physics
- Logic gates
- Which logic gate is given in the figure?
- JEE Main - 2026
- Physics
- Logic gates
- In the figure the LED will glow for input \(A,B,C,D\) (\(0\) is low potential and \(1\) is high potential).
- JEE Main - 2026
- Physics
- Logic gates
Which of the following circuits has the same output as that of the given circuit?
- JEE Main - 2025
- Physics
- Logic gates
View More Questions
Questions Asked in GATE DA exam
- Rishi and Swathi are students of Class 5. Pavan and Tanvi are students of Class 4. Rishi and Pavan are boys. Swathi and Tanvi are girls. These four students played a total of three games of chess. The games were played one after another. A player who lost a game did not participate in any more games. It was observed that:
1. The first game was the only game where two students of the same class played against each other.
2. The students of Class 5 won more games than the students of Class 4.
3. The boys won 2 games and the girl won 1 game.
The student who did not lose any game is:- GATE DA - 2026
- Logical Reasoning
Consider two distinct positive numbers \( m, n \) with \( m > n \). Let \[ x = n^{\log_n m}, \quad y = m^{\log_m n}. \] The relation between \( x \) and \( y \) is -
- GATE DA - 2026
- Logarithms
- Consider the supervised learning task. The objective function being minimized is \( f(w) = w \cdot x \), where \( w \in \mathbb{R} \) is the parameter. Stochastic Gradient Descent with learning rate of 0.10. Let \( w = 10.00 \) be the \( i^{\text{th}} \) iteration (\( w_i \)). The value of \( w \) at the end of iteration \( (i+1) \) is if \( x = 10 \) ________. Round off 2 decimal.
- GATE DA - 2026
- Machine Learning
- The value of \[ \sum_{j=0}^{\infty} \sum_{i=1}^{\infty} 2^{-j} 3^{-i} \] is ________.
- GATE DA - 2026
- Sequences and Series
- Consider the Ridge LRR is being used to learn prediction function \( y_{\text{pred}} = w^T x \) where \( w, x \in \mathbb{R}^2 \) & mean absolute error (MAE) is used to measure the prediction error. A weight of 0.20 is associated with the regularizer. At an intermediate step of training process assume that the parameter \( w = [-3.00, 4.00]^T \). In the next step for the I/P \( x = [1.00, 2.00]^T \), the predicted value of \( y \) is noted. Let the relation b/w \( x = [x_1, x_2]^T \) & the true value of \( y \) be \( y_{\text{true}} = x_1 + x_2 \). The value of the overall regularized loss for instance is __________ (upto 2 decimal).
- GATE DA - 2026
- Machine Learning
View More Questions