Which of the following statements is a tautology?
\( \big((\sim p)\lor q\big)\Rightarrow p \)
\( p\Rightarrow \big((\sim p)\lor q\big) \)
\( \big((\sim p)\lor q\big)\Rightarrow q \)
\( q\Rightarrow \big((\sim p)\lor q\big) \)
Let \( \Delta \in \{\land, \lor, \Rightarrow, \Leftrightarrow\} \) be such that \( (p \land q)\ \Delta\ \big((p \lor q) \Rightarrow q\big) \) is a tautology. Then \( \Delta \) is equal to:
Negation of the Boolean expression \( p \Leftrightarrow (q \Rightarrow p) \) is:
Let the operation \(\ast, \odot \in \{\land, \lor\}\). If \( (p * q)\\odot\ (p \odot \sim q) \) is a tautology, then the ordered pair \( (*,\ \odot) \) is:
Let \( p \): Ramesh listens to music \( q \): Ramesh is out of his village \( r \): It is Sunday \( s \): It is Saturday Then the statement “Ramesh listens to music only if he is in his village and it is Sunday or Saturday” can be expressed as:
\( (p \lor r) \Leftrightarrow \big(p \land (\sim q)\big) \) is equivalent to \( (\sim p) \) when \( r \) is:
Consider the following statements:\( A \) : Rishi is a judge.\( B \) : Rishi is honest.\( C \) : Rishi is not arrogant.The negation of the statement “if Rishi is a judge and he is not arrogant, then he is honest” is:
The negation of the Boolean expression \( \big((\sim q)\land p\big)\Rightarrow \big((\sim p)\lor q\big) \) is logically equivalent to:
Negation of the Boolean statement \( (p \vee q) \Rightarrow ((\sim r) \vee p) \) is equivalent to:
Let \( \Delta, \nabla \in \{\wedge, \vee\} \) be such that \( p \nabla q \Rightarrow ((p \Delta q)\nabla r) \) is a tautology. Then \( (p \nabla q)\Delta r \) is logically equivalent to:
The number of choices for $\Delta \in \{\wedge, \vee, \Rightarrow, \Leftrightarrow\}$, such that $(p \Delta q) \Rightarrow ((p \Delta \sim q) \vee ((\sim p) \Delta q))$ is a tautology, is:
The mean and variance of the data \( 4, 5, 6, 6, 7, 8, x, y \), where \( x < y \), are \( 6 \) and \( \frac{9}{4} \) respectively. Then \( x^4 + y^2 \) is equal to
Let $r \in \{p, q, \sim p, \sim q\}$ be such that the logical statement $r \vee (\sim p) \Rightarrow (p \wedge q) \vee r$ is a tautology. Then $r$ is equal to:
The statement \( \big(\sim(p \Leftrightarrow\sim q)\big)\land q \) is:
Let \( p,q,r \) be three logical statements. Consider the compound statements $S_1 : ((\sim p) \vee q) \vee ((\sim p) \vee r)$ and $S_2 : p \rightarrow (q \vee r)$Then, which of the following is NOT true?
