Z transform is a non-linear operation.

.

ROC include the unit circle hence it is stable.

.

In transfer function, no of zeros = no. of poles, hence it is all pass fliter.

For linearity

y₁[n] = x₁[n] + x₂[n - 10] ...(1)

y₂ = x₂[n] + x₂[n - 10] ...(2)

y₁[n] = x₁[n] + x₂[n] + x₂[n - 10] + x₂[n - 10] ...(3)

Now find y₁[n] + y₂[n]

Corresponding to x₁[n] + x₂[n]

It is same as equation (3) hence linear.

But in part (c) y[n] = x²[n]

? y₁[n] = x²₁[n], y₂[n] = x²₂[n]? y₁[n] + y₂[n] = x²₂[n]...3

But y₁[n] + y₂[n]

Corresponing x₁[n] + x₂[n] is y₁[n] + y₂[n] = {x₁[n] + x₂[n]}²

= x₁²[n] + x₂²[n] + 2x₁[n] x₂[n]....4

Equations (3) and (4) are not same hence not linear.

u_k + 0.5 y_{k - 1} - 0.25 y_{k - 2} = y_k.

.

.

By duality Property.

.