Two equations in a block of table

Question

I make a table, 2*2，

\begin{center}
\begin{table}[H]
\centering
\begin{tabular}{|c|c|} \hline

$F=ax$ &  \nabla F =a $         \\ \hline
$G=a+b\\ K=a+c$ &  $ H=c+d$

\end{tabular}
\end{table}
\end{center}

The above code generates mistakes. In 2*1 block I want a 2 line equation. So I try \align,\equation and so on. For example,

\begin{center}
\begin{table}[H]
\centering
\begin{tabular}{|c|c|} \hline

$F=ax$ &  \nabla F =a $         \\ \hline
\begin{align*}G&=a+b\\ K&=a+c\end{align*} &  $ H=c+d$

\end{tabular}
\end{table}
\end{center}

But this makes mistake too. So I have to use $$, but I do not know how to write two equation in $$.

Thank you. I am looking forward to your answer.

By the way, even this makes mistakes.

\begin{center}
\begin{table}[H]
\centering
\begin{tabular}{|c|c|} \hline

$F=ax$ &  \nabla F =a $         \\ \hline
\begin{equation*}G=a+b \end{equation*} &  $ H=c+d$

\end{tabular}
\end{table}
\end{center}

Answer 1

I'm sorry, but I can not recognize your handwriting.

Do you mean something like this?

Code:

\documentclass{article}
\begin{document}
\begin{table}
\centering
$ \begin{array}{|c|c|}
\hline
F(x) & G(x) \\
\hline
F(N) & G(N) \\
N=a & N=a \\
\hline
\end{array} $
\end{table}
\end{document} 

Answer 2

I'm not sure I understand fully what you're trying to achieve since you refer to a 2x2 table in one place and a 2x1 table in another.

Since the contents of the table would appear to be math expressions, you may want to use an array instead of a tabular environment. For a 2x2 table you could proceed along the following lines:

\documentclass{article}
\begin{document}
\begin{table}
\centering
$ \begin{array}{|c|c|} 
\hline
F & \nabla F \\ 
\hline
G & \nabla G \\
\hline
\end{array} $
\end{table}
\end{document}