Implementation of AND Gate from NAND Gate

The AND gate can be implemented using only NAND gates because NAND gates are functionally complete, meaning they can be used to construct any other basic logic gate.

Truth Table for AND Gate

The AND gate outputs 1 only if both inputs are 1. Its truth table is:

Using NAND Gates to Build AND Gate

The NAND gate operates as follows:

Output of NAND=¬(A∧B)\text{Output of NAND} = \neg(A \land B)

This is the inverse of an AND gate. To construct an AND gate, we need to invert the output of a NAND gate.

Steps to Create AND Gate Using NAND Gates

1. Use a single NAND gate to calculate ¬(A∧B)\neg(A \land B).
2. Pass the output of the first NAND gate through another NAND gate where both inputs are connected to the same output. This acts as a NOT gate to invert the signal.

Circuit Diagram

1. First NAND Gate:
   • Inputs: AA, BB
   • Output: ¬(A∧B)\neg(A \land B)
2. Second NAND Gate (as NOT Gate):
   • Inputs: Output of the first NAND gate (connected to both inputs)
   • Output: A∧BA \land B

Logical Expression

1. First NAND gate: Q=¬(A∧B)Q = \neg(A \land B)
2. Second NAND gate: Z=¬(Q∧Q)=¬(¬(A∧B))=A∧BZ = \neg(Q \land Q) = \neg(\neg(A \land B)) = A \land B

Implementation in Python

Here is a simple Python implementation of an AND gate using NAND logic:

# NAND gate function
def NAND(a, b):
    return not (a and b)

# AND gate using NAND gates
def AND(a, b):
    nand_output = NAND(a, b)   # First NAND gate
    and_output = NAND(nand_output, nand_output)  # Second NAND gate
    return and_output

# Test the AND gate
inputs = [(0, 0), (0, 1), (1, 0), (1, 1)]
print("A  B  A AND B")
for a, b in inputs:
    print(f"{a}  {b}    {int(AND(a, b))}")

Output:

A  B  A AND B
0  0    0
0  1    0
1  0    0
1  1    1

Conclusion

Using just two NAND gates, you can implement the functionality of an AND gate. This demonstrates the versatility of the NAND gate in digital logic design.