Answer
$\begin{array}{lllllll}
p & q & \sim p & \sim q & p\vee q & \sim p\vee\sim q & (p\vee q)\wedge(\sim p\vee\sim q)\\
\hline T & T & F & F & T & F & F\\
T & F & F & T & T & T & T\\
F & T & T & F & T & T & T\\
F & F & T & T & F & T & F
\end{array}$
Work Step by Step
Set up a truth table for two inputs, p and q:
$\begin{array}{llll}
p & q & ... & ...\\
\hline T & T & & \\
T & F & & \\
F & T & & \\
F & F & &
\end{array}$
In columns 3 and 4, use the negation table for $\sim p$ and $\sim q$
$\begin{array}{llllll}
p & q & \sim p & \sim q & & \\
\hline T & T & F & F & & \\
T & F & F & T & & \\
F & T & T & F & & \\
F & F & T & T & &
\end{array}$
Next column: $p\vee q$
(disjunction, inputs: columns 1 and 2)
$\begin{array}{llllll}
p & q & \sim p & \sim q & p\vee q & \\
\hline T & T & F & F & T & \\
T & F & F & T & T & \\
F & T & T & F & T & \\
F & F & T & T & F &
\end{array}$
Next column: $\sim p\vee\sim q$
(disjunction, inputs: columns 3 and 4)
$\begin{array}{llllll}
p & q & \sim p & \sim q & p\vee q & \sim p\vee\sim q\\
\hline T & T & F & F & T & F\\
T & F & F & T & T & T\\
F & T & T & F & T & T\\
F & F & T & T & F & T
\end{array}$
Final column: $(p\vee q)\wedge(\sim p\vee\sim q)$
(conjunction, inputs: columns $5$ and $6$)
$\begin{array}{lllllll}
p & q & \sim p & \sim q & p\vee q & \sim p\vee\sim q & (p\vee q)\wedge(\sim p\vee\sim q)\\
\hline T & T & F & F & T & F & F\\
T & F & F & T & T & T & T\\
F & T & T & F & T & T & T\\
F & F & T & T & F & T & F
\end{array}$
