Question

Explain full adder with its logic diagram & truth table.

Answer 1

A **full adder** is a digital circuit that performs addition of binary numbers. It takes three inputs: two significant bits (A and B) and a carry-in bit (C_in) from a previous less significant stage. The full adder produces two outputs: a sum (S) and a carry-out (C_out) to the next higher significant stage.

### Logic Diagram

The logic diagram of a full adder can be represented using basic logic gates: AND, OR, and XOR gates.


### Explanation of the Logic Diagram:

- **Inputs:**
  - A: First bit to be added.
  - B: Second bit to be added.
  - C_in: Carry input from the previous addition.

- **Outputs:**
  - S: Sum output.
  - C_out: Carry output to the next addition.

- **Gates Used:**
  - **XOR Gates**: Used to compute the sum.
  - **AND Gates**: Used to determine if a carry should be generated.
  - **OR Gate**: Combines the carry outputs from the AND gates.

### Truth Table

The truth table for a full adder is as follows:

| A | B | C_in | S (Sum) | C_out (Carry) |
|---|---|-------|---------|----------------|
| 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        |

### Output Derivation

1. **Sum (S)**:
   - The sum output can be derived from the following logic:
   \[
   S = A \oplus B \oplus C_{in}
   \]
   This means the sum is the XOR of the two inputs and the carry-in.

2. **Carry-out (C_out)**:
   - The carry-out can be derived as follows:
   \[
   C_{out} = (A \land B) \lor (C_{in} \land (A \oplus B))
   \]
   This means the carry-out occurs if either:
   - Both A and B are 1 (producing a carry), or
   - One of A or B is 1, and the carry-in is also 1.

### Applications

Full adders are widely used in digital circuits, particularly in:
- Arithmetic logic units (ALUs).
- Building more complex adders like ripple-carry adders and carry-lookahead adders.
- Digital signal processing.

By understanding the workings of a full adder, you can grasp how binary addition is performed at the hardware level, forming the basis for more complex arithmetic operations in computing systems.

Answer 2

A **Full Adder** is a digital circuit used for adding three binary digits, where the addition might produce a carry out. It takes three inputs: two significant bits and a carry-in bit from a previous addition, and produces two outputs: a sum bit and a carry-out bit.

### 1. Logic Diagram

The Full Adder can be implemented using basic logic gates such as XOR, AND, and OR gates. Here’s the logic diagram for a Full Adder:

```
        A  ----
               |
               |     XOR ---- SUM
        B  ----|----|
               |     |
               |     XOR ----
               |
        CIN ----|
               |        AND ----
               |              |
        A ----|              |--- OR ---- COUT
               |              |
        B ----|              |
                              AND
```

In the diagram:
- **A** and **B** are the two significant bits.
- **CIN** is the carry-in bit from the previous less significant digit.
- **SUM** is the output sum.
- **COUT** is the carry-out bit, which is passed to the next more significant digit.

### 2. Truth Table

The truth table for the Full Adder provides a detailed view of the output values for each combination of inputs:

| 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  |

### 3. Logic Expressions

The logic expressions for the SUM and COUT can be derived from the truth table:

- **SUM** is computed using the XOR gate:
  
  \[
  \text{SUM} = A \oplus B \oplus CIN
  \]

- **COUT** is computed using a combination of AND and OR gates:
  
  \[
  \text{COUT} = (A \land B) \lor (B \land CIN) \lor (CIN \land A)
  \]

### Summary

The Full Adder is a fundamental building block in digital electronics, especially in arithmetic logic units (ALUs) for binary addition. Its design allows for the extension to multiple-bit binary addition by chaining Full Adders together, with the carry-out of one serving as the carry-in for the next more significant bit.