Design half adder using NAND gates. Also draw the truth table.

Q.) Design half adder using NAND gates. Also draw the truth table.
Subject: Digital Systems

A half adder is a type of adder, an electronic device that performs the addition of numbers. The half adder is able to add two single binary digits and it provides the output as well as a carry value. It has two inputs, known as the augend and addend bits, and two outputs, the sum (S) and carry (C) bits.

The half adder can be implemented using NAND gates only. Here is the step by step approach:

First, we need to understand the Boolean expression for Sum and Carry. The Sum (S) is given by A XOR B and the Carry (C) is given by A AND B.
The XOR gate can be implemented using NAND gates as follows:

A XOR B = (A NAND (A NAND B)) NAND (B NAND (A NAND B))
The AND gate can be implemented using NAND gates as follows:

A AND B = (A NAND B) NAND (A NAND B)
Now, we can design the half adder using these expressions.

Here is the circuit diagram:

    A ----&gt; | NAND |-------------------&gt; | NAND |-------------------&gt; S
            |  1   |                     |  3   |
            |      |                     |      |
    B ----&gt; | NAND |----&gt; | NAND | &lt;-----| NAND |
            |  2   |      |  4   |       |  5   |
            |      |      |      |       |      |
    B ----&gt; | NAND | &lt;----| NAND |       | NAND |
            |  6   |      |  7   |       |  8   |
            |      |      |      |       |      |
    A ----&gt; | NAND |-------------------&gt; | NAND |-------------------&gt; C

In the above diagram, gates 1 and 2 implement A NAND B, gates 4 and 7 implement A NAND (A NAND B) and B NAND (A NAND B) respectively, and gate 3 implements the XOR operation. Gates 5 and 8 implement the AND operation.

The truth table for the half adder using NAND gates is as follows:

A	B	S = (A NAND (A NAND B)) NAND (B NAND (A NAND B))	C = (A NAND B) NAND (A NAND B)
0	0	0	0
0	1	1	0
1	0	1	0
1	1	0	1

In the truth table, S represents the sum and C represents the carry. The sum is 1 when one of the inputs (A or B) is 1, and the carry is 1 only when both inputs (A and B) are 1.