Statement I: The area by region $|x+y|+|x-y|\le 2$  is 8 sq units 
Statement II: Area enclosed by region $|x+y|+|x-y|\le 2$ is symmetric about x-axis
 

As the area enclosed by $\left|x\right|+\left|y\right|\le a$  is the area of square (i.e. $2a^2$  )

Area enclosed by $|x+y|+|x-y|\le 2$  is area of square shown as $area=4(\frac{1}{2}\times 2\times 2)=8squnits$
Also the area enclosed by $|x+y|+|x-y|\le 2$  is symmetric about x-axis $y=xandy=-x$
y-axis,  
Both the statements are true but statement II is not the correct explanation of Statement I