How to justify that M_AC=0 for a symmetric airfoil? I learnt that for a symmetric airfoil, the pitch moment around the aerodynamic center should be null:

$$M_{AC}=0$$
$\hspace{11pt}$ How could I intuitively explain this statement?

I also learnt that for an airfoil with positive curvature, $M_{AC}&lt;0$ and  for an airfoil with negative curvature, $M_{AC}&gt;0$. I'm new at my Flight Stability course, and I don't know how to achieve these conclusions. I do know the definition of aerodynamic center, but it isn't enough for me to make such observations.
How could I deduce these statements?
Per definition the moment coefficient in the Aerodynamic Centre is independent of the angle of attack. 
Since the airfoil is symmetrical, the moment around the a.c., $M_{a.c.}$ for an angle of attack $\alpha $ will be of equal magnitude and opposite sign at angle of attack $-\alpha $: 
$M_{a.c.}(\alpha)=-M_{a.c.}(-\alpha)$


The two statements can only hold if $M_{a.c.} = 0$