What is the identity element for the sum (disjunction) in Boolean algebra of logical statements?

Negation of the statement: $\sqrt{5}$ is an integer or $5$ is irrational is

For any three simple statements $p, q, r$,the statement $(p \wedge q) \vee (q \wedge r)$ is true if and only if:

The negation of the compound statement $\sim p \vee (p \vee (\sim q))$ 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

Consider
Statement-$1$: $(p \wedge \sim q) \wedge (\sim p \wedge q)$ is a fallacy.
Statement-$2$: $(p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p)$ is a tautology.