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The statement \( \big(p \land (\sim q)\big) \lor \big((\sim p)\land q\big) \lor \big((\sim p)\land (\sim q)\big) \) is equivalent to ________.

The statement \( (p \land (\sim q)) \Rightarrow (p \Rightarrow (\sim q)) \) is:

The negation of the statement \( (p \vee q)\wedge(q \vee (\sim r)) \) is:

The compound statement \( (\sim(P \land Q)) \lor ((\sim P) \land Q) \Rightarrow ((\sim P) \land (\sim Q)) \) is equivalent to:

The number of values of $r \in \{p, q, \sim p, \sim q\}$ for which $((p \wedge q) \Rightarrow (r \vee q)) \wedge ((p \wedge r) \Rightarrow q)$ is a tautology, is:

The statement \( B \Rightarrow ((\sim A) \lor B) \) is equivalent to:

Among the two statements
\( (S1): (p \Rightarrow q) \land \left(p \land (\sim q)\right) \) is a contradiction and
\( (S2): (p \land q) \lor \left((\sim p) \land q\right) \lor \left(p \land (\sim q)\right) \lor \left((\sim p) \land (\sim q)\right) \) is a tautology.

For the system of linear equations
\( 2x - y + 3z = 5 \)
\( 3x + 2y - z = 7 \)
\( 4x + 5y + \alpha z = \beta \),
Which of the following is NOT correct?

Negation of \( p \wedge (q \wedge \sim(p \wedge q)) \) is:

Among the statements:
(\( S1 \)): \( (p \Rightarrow q) \vee ((\sim p) \wedge q) \) is a tautology.
(\( S2 \)): \( (q \Rightarrow p) \Rightarrow ((\sim p) \wedge q) \) is a contradiction.

Negation of \( (p \Rightarrow q) \Rightarrow (q \Rightarrow p) \) is:

The converse of \( ((\sim p)\wedge q) \Rightarrow r \) is:

Among the statements:
\( (S1) \; ((p \lor q) \Rightarrow r) \Leftrightarrow (p \Rightarrow r) \)
\( (S2) \; ((p \lor q) \Rightarrow r) \Leftrightarrow ((p \Rightarrow r) \lor (q \Rightarrow r)) \)

Let \( \Delta,\nabla \in \{\land,\lor\} \) be such that \( (p \to q)\,\Delta\,(p \nabla q) \) is a tautology. Then

Which of the following statements is a tautology?

The negation of \( \big(p\land(\sim q)\big)\lor(\sim p) \) is equivalent to:

The negation of the expression \( q \vee \big((\sim q) \wedge p\big) \) is equivalent to: