Answer
$\begin{array}{llllll}
p & q & \sim q & p\vee\sim q & p\vee q & ( p\vee\sim q)\wedge(p\vee q)\\
\hline T & T & F & T & T & T\\
T & F & T & T & T & T\\
F & T & F & F & T & F\\
F & F & T & T & F & F
\end{array}$
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 the next column (3rd) , use the negation table for $\sim q$
$\begin{array}{lllll}
p & q & \sim q & & \\
\hline T & T & F & & \\
T & F & T & & \\
F & T & F & & \\
F & F & T & &
\end{array}$
Next column: $p\vee\sim q$
(disjunction, inputs: columns 1 and 3)
$\begin{array}{lllll}
p & q & \sim q & p\vee\sim q & \\
\hline T & T & F & T & \\
T & F & T & T & \\
F & T & F & F & \\
F & F & T & T &
\end{array}$
Next column: $p\vee q$
(disjunction, inputs: columns 1 and 2)
$\begin{array}{lllll}
p & q & \sim q & p\vee\sim q & p\vee q\\
\hline T & T & F & T & T\\
T & F & T & T & T\\
F & T & F & F & T\\
F & F & T & T & F
\end{array}$
Final column: $( p\vee\sim q)\wedge(p\vee q)$
(conjunction, inputs: columns 4 and 5)
$\begin{array}{llllll}
p & q & \sim q & p\vee\sim q & p\vee q & ( p\vee\sim q)\wedge(p\vee q)\\
\hline T & T & F & T & T & T\\
T & F & T & T & T & T\\
F & T & F & F & T & F\\
F & F & T & T & F & F
\end{array}$
