Verifying Truth Tables would someone please verify the following truth table? If there is something innacurate, would someone please explain why?
Thank You!
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\color{black}{L} & \color{black}{G} & \color{black}{H} & \lnot H & (G\lor\lnot H)& \lnot(G\land L) &(G\lor \lnot H )\land \lnot(G\land L) \\ \hline
\color{}{T} & \color{}{T} & \color{}{T} &   \color{}{F}     & \color{}{T}& \color{}{F}&\color{}{F} \\  \hline
\color{black}{T} & \color{black}{T} & \color{black}{F} &   \color{black}{T}     & \color{black}{T}&  \color{black}{F}&\color{black}{F} \\ \hline
\color{black}{T} & \color{black}{F} & \color{black}{T} &   \color{black}{F}     & \color{black}{F}& \color{black}{T} &\color{black}{F} \\ \hline
\color{black}{T} & \color{black}{F} & \color{black}{F} &   \color{black}{T}     & \color{black}{T}& \color{black}{T}&\color{black}{T} \\ \hline
\color{black}{F} & \color{black}{T} & \color{black}{T} &   \color{black}{F}     & \color{black}{T}& \color{black}{T}&\color{black}{T} \\ \hline
\color{black}{F} & \color{black}{T} & \color{black}{F} &   \color{black}{T}     & \color{black}{T}& \color{black}{T}&\color{black}{T} \\ \hline
\color{black}{F} & \color{black}{F} & \color{black}{T} &   \color{black}{F}     & \color{black}{F}& \color{black}{T}&\color{black}{T} \\ \hline
\color{black}{F} & \color{black}{F} & \color{black}{F} &   \color{black}{T}     & \color{black}{T}& \color{black}{T}&\color{black}{T} \\ \hline
\end{array}$$
 A: Since you provided a nicely formatted table, I am happy to provide my own solutions. I marked any mistake, and entries that - while being correct - were affected by previous mistakes, in red and hopefully caught all of them (;
\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\color{black}{L} & \color{black}{G} & \color{black}{H} & \lnot H & (G\lor\lnot H)& \lnot(G\land L) &(G\lor \lnot H )\land \lnot(G\land L) \\ \hline
\color{}{T} & \color{}{T} & \color{}{T} &   \color{}{F}     & \color{}{T}& \color{}{F}&\color{}{F} \\  \hline
\color{black}{T} & \color{black}{T} & \color{black}{F} &   \color{black}{T}     & \color{black}{T}&  \color{black}{F}&\color{black}{F} \\ \hline
\color{black}{T} & \color{black}{F} & \color{black}{T} &   \color{black}{F}     & \color{black}{F}& \color{black}{T} &\color{black}{F} \\ \hline
\color{black}{T} & \color{black}{F} & \color{black}{F} &   \color{black}{T}     & \color{black}{T}& \color{black}{T}&\color{black}{T} \\ \hline
\color{black}{F} & \color{black}{T} & \color{black}{T} &   \color{black}{F}     & \color{black}{T}& \color{black}{T}&\color{black}{T} \\ \hline
\color{black}{F} & \color{black}{T} & \color{black}{F} &   \color{black}{T}     & \color{black}{T}& \color{black}{T}&\color{black}{T} \\ \hline
\color{black}{F} & \color{black}{F} & \color{black}{T} &   \color{black}{F}     & \color{black}{F}& \color{black}{T}&\color{red}{F} \\ \hline
\color{black}{F} & \color{black}{F} & \color{black}{F} &   \color{red}{T}     & \color{red}{T}& \color{black}{T}&\color{red}{T} \\ \hline
\end{array}

edit: Since you also asked for an explanation, let us enumerate the mistakes from top to bottom, left to right:


*

*A conjunction $\phi \wedge \psi$ is true iff both $\phi$ and $\psi$ are true. Hence, since $(G \vee \neg H)$ is false $(G \vee \neg H) \wedge \neg(G \wedge L)$ must also be false.

*Since $H$ is  false, $\neg H$ is true.

*Your post already, correctly, declared $(G \vee \neg H)$ to be true. However,  since you accidentally declared $\neg H$ to be false, this wasn't justified.

*I only marked this as a mistake, because it depended on item 3. which in turn depended on 2.

