Mathematical Reasoning

The statement $p \to \left( {q \to p} \right)$ is equivalent to

Statement-1 : $\sim \left( {p \leftrightarrow \sim q} \right)$ is equivalent to ${p \leftrightarrow q}$. Statement-2 : $\sim \left( {p \leftrightarrow \sim q} \right)$ is a tautology.

Consider : Statement − I : $\left( {p \wedge \sim q} \right) \wedge \left( { \sim p \wedge q} \right)$ is a fallacy. Statement − II :$\left( {p \to q} \right) \leftrightarrow \left( { \sim q \to \sim p} \right)$ is a tautology.

The statement $\sim \left( {p \leftrightarrow \sim q} \right)$ is :

The following statement $\left( {p \to q} \right) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]$ is :

If the truth value of the statement p $\to$ (~q $\vee$ r) is false (F), then the truth values of the statements p, q, r are respectively :

The negation of the Boolean expression ~ s $\vee$ (~r $\wedge$ s) is equivalent to :

Which one of the following Boolean expressions is a tautology?

If p $\Rightarrow$ (q $\vee$ r) is false, then the truth values of p, q, r are respectively :-

For any two statements p and q, the negation of the expression p $\vee$ (~p $\wedge$ q) is :