If truth values of statements \(p, q\) are true, and \(r\), \(s\) are false, then the truth values of the following statement patterns are respectively
\(\mathrm{a}: \sim(\mathrm{p} \wedge \sim \mathrm{r}) \vee(\sim \mathrm{q} \vee \mathrm{s})\)
\(\mathrm{b}:(\sim \mathrm{q} \wedge \sim \mathrm{r}) \leftrightarrow(\mathrm{p} \vee \mathrm{s})\)
\(c:(\sim p \vee q) \rightarrow(r \wedge \sim s)\)

a.
\(\begin{aligned}
& \sim(\mathrm{p} \wedge \sim \mathrm{r}) \vee(\sim \mathrm{q} \vee \mathrm{s}) \\
& \equiv \sim(\mathrm{T} \wedge \sim \mathrm{F}) \vee(\sim \mathrm{T} \vee \mathrm{F}) \\
& \equiv(\mathrm{F} \vee \mathrm{F}) \vee(\mathrm{F} \vee \mathrm{F}) \\
& \equiv \mathrm{F} \vee \mathrm{F} \\
& \equiv \mathrm{F}
\end{aligned}\)
b.
\(\begin{aligned}
& (\sim \mathrm{q} \wedge \sim \mathrm{r}) \leftrightarrow(\mathrm{p} \vee \mathrm{s}) \\
& \equiv(\sim \mathrm{T} \wedge \sim \mathrm{F}) \leftrightarrow(\mathrm{T} \vee \mathrm{F}) \\
& \equiv \mathrm{F} \leftrightarrow \mathrm{T} \\
& \equiv \mathrm{F}
\end{aligned}\)
c.
\(\begin{aligned}
& (\sim p \vee q) \rightarrow(r \wedge \sim s) \\
& \equiv(\sim T \vee T) \rightarrow(F \wedge \sim F) \\
& \equiv(F \vee T) \rightarrow(F \wedge T) \\
& \equiv T \rightarrow F \\
& \equiv F
\end{aligned}\)