Design and draw a full adder circuit.

A full adder is a digital circuit that performs addition of three binary bits. It has three inputs: A, B, and Cin (carry in), and two outputs: Sum and Cout (carry out). The full adder can be used in constructing arithmetic circuits like adders and subtractors.

Step 1: Understand the Truth Table

To design a full adder, we first need to understand its truth table, which defines the relationship between the inputs and outputs.

A	B	Cin	Sum	Cout
0	0	0	0	0
0	0	1	1	0
0	1	0	1	0
0	1	1	0	1
1	0	0	1	0
1	0	1	0	1
1	1	0	0	1
1	1	1	1	1

Step 2: Derive the Logic Expressions

From the truth table, we can derive the logic expressions for the Sum and Cout using Karnaugh maps or Boolean algebra.

For Sum: [ \text{Sum} = A \oplus B \oplus \text{Cin} ]

For Cout: [ \text{Cout} = (A \cdot B) + (B \cdot \text{Cin}) + (A \cdot \text{Cin}) ]

Step 3: Implement the Logic Circuit

Using the derived logic expressions, we can now implement the full adder circuit using logic gates.

1. The Sum output can be implemented using two XOR gates.
2. The Cout output can be implemented using AND gates and an OR gate.

Step 4: Draw the Full Adder Circuit

Here is the schematic of a full adder circuit:

       A ───┬───┐
            │   │
            XOR ───┐
            │     │
       B ───┴───┐ │
                XOR ─── Sum
       Cin ─────┤ │
                │ │
                AND ──┐
                │     │
       A ───────┤     │
                │     │
                AND ──┼─── Cout
                │     │
       B ───────┤     │
                │     │
                AND ──┘
                │
       Cin ─────┘

Step 5: Verify the Circuit

To verify the full adder circuit, you can simulate it using software like Logisim, or you can build it on a breadboard using physical logic gates and test it with different combinations of input values to ensure it matches the truth table.

Example

Let's consider an example where A=1, B=1, and Cin=0.

According to the truth table:

• The Sum should be 0 (since 1 + 1 = 10 in binary, the sum is the least significant bit).
• The Cout should be 1 (since 1 + 1 = 10 in binary, the carry is the most significant bit).

Using the logic expressions:

• Sum = A ⊕ B ⊕ Cin = 1 ⊕ 1 ⊕ 0 = 0
• Cout = (A ⋅ B) + (B ⋅ Cin) + (A ⋅ Cin) = (1 ⋅ 1) + (1 ⋅ 0) + (1 ⋅ 0) = 1 + 0 + 0 = 1

The example confirms that the full adder circuit works as expected for this set of inputs.