Digital Logic and Microprocessors

By De Morgan's Law, inverting the inputs of an AND gate and then inverting the output is equivalent to an OR gate (A' * B')' = A + B.

Both NAND and NOR gates are considered universal gates because any other logic gate can be implemented using only one type of these gates. Between the options provided, NOR is a universal gate.

To perform an AND operation using NAND gates, you need two: the first NAND gate performs the AND operation (with inputs X and Y), and the second NAND gate acts as an inverter to restore the AND logic.

Simplify (X + Y + XY)(X + Z). Since X + XY = X, the first term becomes (X + Y). Then (X + Y)(X + Z) = X + XZ + YX + YZ = X + XZ + XY + YZ = X(1 + Z + Y) + YZ = X + YZ.

NAND gates are universal gates, meaning any Boolean expression can be implemented using only NAND gates.

A toggle operation (like an XOR) cannot be performed with a single AND gate because an AND gate is not universal and cannot implement the XOR function on its own.

The condition f(x, y, z) = f(x', y', z') means the function is invariant under complementation of all inputs. There are 2^3 = 8 possible input combinations, which can be grouped into 4 pairs of (x, y, z) and (x', y', z'). For each pair, the function can take 2 values (0 or 1), leading to 2^4 = 16 possible functions.

Option D represents the Commutative Law of Boolean algebra, which states that x + y = y + x. Option A is incorrect (x.x = x), Option B is incorrect (distributive law is x + yz = (x + y)(x + z)), and Option C is incorrect (x + xz = x is actually the Absorption Law, but the question asks for a general identity).

In a standard SR flip-flop, the state S = 1, R = 1 is considered invalid or forbidden because it leads to an indeterminate state where both outputs Q and Q' attempt to go to 0.

Placing an inverter between the S and R inputs of an SR flip-flop ensures that S and R are always complements of each other. This prevents the forbidden state and creates a D flip-flop, where the output follows the input.

A flip-flop is a bistable multivibrator, which is the fundamental building block of sequential logic circuits, capable of storing exactly one bit of binary information.

To divide a frequency by N, you need n flip-flops such that 2^n >= N. For N = 64, 2^6 = 64, so 6 flip-flops are required.

A multiplexer (MUX) is a combinational circuit that selects one of several input lines and routes it to a single output line based on select signals.

2's complement of a number is obtained by adding 1 to the 1's complement of that number.

2's complement of a number is obtained by adding 1 to the 1's complement of that number.