Which one of the following is NOT a valid identity?

Consider Z = X - Y, where X, Y and Z are all in sign-magnitude form. X and Y are each represented in n bits. To avoid overflow, the representation of Z would require a minimum of:

What is the minimum number of 2-input NOR gates required to implement a 4-variable function function expressed in sum-of-minterms form as

f = Σ(0, 2, 5, 7, 8, 10, 13, 15)?

Assume that all the inputs and their complements are available.

Consider three 4-variable functions f₁, f₂ and f₃, which are expressed in sum-of-minterms as

f₁ = Σ(0, 2, 5, 8, 14),

f₂ = Σ(2, 3, 6, 8, 14, 15),

f₃ = Σ(2, 7, 11, 14)

For the following circuit with one AND gate and one XOR gate, the output function f can be expressed as: