1. Which of the following justifies the linearity property of z-transform?[x(n)↔X(z)].a) x(n)+y(n) ↔ X(z)Y(z)b) x(n)+y(n) ↔ X(z)+Y(z)c) x(n)y(n) ↔ X(z)+Y(z)d) x(n)y(n) ↔ X(z)Y(z)

To justify the linearity property of the z-transform, we need to find the correct option among the given choices.

The linearity property states that if x(n) ↔ X(z) and y(n) ↔ Y(z), then for any constants a and b, ax(n) + by(n) ↔ aX(z) + bY(z).

Let's evaluate each option:

a) x(n) + y(n) ↔ X(z)Y(z) This option does not match the linearity property because the z-transform of the sum of two signals should be the sum of their individual z-transforms, not their product.

b) x(n) + y(n) ↔ X(z) + Y(z) This option matches the linearity property. The z-transform of the sum of two signals is indeed the sum of their individual z-transforms.

c) x(n)y(n) ↔ X(z) + Y(z) This option does not match the linearity property because the z-transform of the product of two signals should be the convolution of their individual z-transforms, not their sum.

d) x(n)y(n) ↔ X(z)Y(z) This option does not match the linearity property because the z-transform of the product of two signals should be the convolution of their individual z-transforms, not their product.

Therefore, the correct option that justifies the linearity property of the z-transform is b) x(n) + y(n) ↔ X(z) + Y(z).