How to justify that M_AC=0 for a symmetric airfoil?
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 $M_{a.c.} = 0$