19. NOR Logic Synthesis

Solution Explanation

From the behavioral table, the circuit must generate AND, OR, and NOT A outputs using only NOR gates.

AND, OR, NOT, NAND, XOR, and XNOR gates are prohibited. Only NOR gates can be used.

1. NOT Using NOR

Required:

Y = A'

Using the Boolean law:

A + A = A

Therefore:

Y = (A + A)'

So, connect A to both inputs of a NOR gate.

2. OR Using NOR

Required:

Y = A + B

Apply double inversion:

Y = (A + B)''       // By law A'' = A

Assume:

Z = (A + B)'        // First NOR gate

Then:

Y = Z'

From Step 1, NOT can be implemented using NOR:

Y = (Z + Z)'

Therefore:

OR Output = ((A + B)' + (A + B)')'

3. AND Using NOR

Required:

Y = A · B

Using De Morgan's theorem:

Y = (A' + B')'

From Step 1:

A' = (A + A)'
B' = (B + B)'

Substituting:

AND Output = ((A + A)' + (B + B)')'

Now the three functions can be implemented as:

Connect the NOR gates according to these derived expressions to generate all three outputs.