The given circuit is equivalent to:

The Boolean expression $(p \wedge q) \Rightarrow ((r \wedge q) \wedge p)$ is equivalent to:

Consider the two statements:
$(S1): (p$ $\rightarrow q) \vee (\sim q$ $\rightarrow p)$ is a tautology.
$(S2): (p \wedge \sim q) \wedge (\sim p \vee q)$ is a fallacy.
Then:

The negation of the statement "For all real numbers $x$ and $y, x+y=y+x$" is

For the statements $p$ and $q$,consider the following compound statements :
$(a)$ $(\sim q \wedge (p$ $\rightarrow q))$ $\rightarrow \sim p$
$(b)$ $((p \vee q) \wedge \sim p) \rightarrow q$
Then which of the following statements is correct?

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