Non invertibility of system $y[n]=x[n]-x[n-1]$ using transform method?
The impulse response of your system is $h[n] = \delta[n] - \delta[n-1]$ and its Z-transform is accordingly

$$H(z)=1-z^{-1}$$

Now, you can evaluate the frequency response of this system:

$$
|H(z)|_{z=\exp(j\omega)}=|1-\exp(j\omega)|
$$

you see, that $H(z)|_{z=\exp(j0)}=0$ (i.e. it vanishes for $\omega=0$). Now, in order for a filter to be invertible, its Z-Transform on the unit circle (i.e. when setting $z=\exp(j\omega)$ must not vanish. If it does not vanish, the inverse is given by $\frac{1}{H(z)}$. If it vanishes, the inverse does not exist.