We know that $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$, and we also know the values $Q$ and $P$ respectively of $q$ and $p$ at a later time step $\Delta t$. How could we prove that the quantities
$$
\begin{align}
Q &= q + {\Delta}t\frac{\partial H}{\partial p}(q,p),\\
P &= p - {\Delta}t\frac{\partial H}{\partial q}(q,p)
\end{align}
$$
are not symplectic, while
$$
\begin{align}
Q &= q - {\Delta}t\frac{\partial H}{\partial p}(q,p),\\
P &= p + {\Delta}t\frac{\partial H}{\partial Q}(Q,p) 
\end{align}
$$
are symplectic?