18. NAND Logic Synthesis

Solution Explanation

From the behavioral table, the circuit must generate AND, OR, and NOT A outputs using only NAND gates.

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

1. NOT Using NAND

Required:

Y = A'

Using the Boolean law:

A · A = A

Therefore:

Y = (A · A)'

So, connect A to both inputs of a NAND gate.

2. AND Using NAND

Required:

Y = A · B

Apply double inversion:

Y = (A · B)''       // By law A'' = A

Assume:

Z = (A · B)'        // First NAND gate

Then:

Y = Z'

From Step 1, NOT can be implemented using NAND:

Y = (Z · Z)'

Therefore:

AND Output = ((A · B)' · (A · B)')'

3. OR Using NAND

Required:

Y = A + B

Using De Morgan's theorem:

Y = (A' · B')'

From Step 1:

A' = (A · A)'
B' = (B · B)'

Substituting:

OR Output = ((A · A)' · (B · B)')'

Now the three functions can be implemented as:

Connect the NAND gates according to these derived expressions to generate all three outputs.