$(p \wedge \sim q) \wedge (\sim p \vee q)$ 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

If $p$: It rains today,$q$: $I$ go to school,$r$: $I$ shall meet my friend,and $s$: $I$ shall go for a movie,then which of the following represents the proposition: "If it does not rain or if $I$ do not go to school,then $I$ shall meet my friend and go for a movie"?

Which of the following statements is correct?
$(a)$ $S_1: (p \wedge q) \equiv \sim(p \rightarrow \sim q)$
$(b)$ $S_2: (p \wedge q) \wedge (\sim p \vee \sim q)$ is a tautology
$(c)$ $S_3: [p \wedge (p$ $\rightarrow \sim q)]$ $\rightarrow q$ is a contradiction
$(d)$ $S_4: p$ $\rightarrow (q$ $\rightarrow p)$ is a contingency

Which of the following sentences are statements? Give reasons for your answer.
All real numbers are complex numbers.

In a Boolean Algebra $B$,for all $x, y \in B$,$x \wedge (x \vee y)$ is equal to