Draw the truth table and logic diagram of full adder.

Q.) Draw the truth table and logic diagram of full adder.

 Subject: Digital Electronics

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: S (sum) and Cout (carry out). The full adder can be used in succession to add binary numbers of any length by chaining the carry output to the carry input of the next full adder in the sequence.

Truth Table

The truth table for a full adder is as follows:

A B Cin S (Sum) Cout (Carry Out)
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

Formulas

The sum (S) and carry out (Cout) can be represented by the following Boolean expressions:

[ S = A \oplus B \oplus Cin ] [ Cout = (A \cdot B) + (B \cdot Cin) + (A \cdot Cin) ]

Where:

  • ( \oplus ) represents the XOR (exclusive OR) operation
  • ( \cdot ) represents the AND operation
  • ( + ) represents the OR operation

Logic Diagram

To draw the logic diagram, we'll use the Boolean expressions to create a circuit with logic gates.

  1. The sum (S) is the result of XORing all three inputs: A, B, and Cin.
  2. The carry out (Cout) is true if any two inputs are true, or all three are true. This can be achieved by ORing the result of three AND gates, each taking two of the inputs.

Here is the logic diagram based on the formulas:

    A -----|     |-----\
           | XOR |      \
    B -----|     |       | XOR |--- S (Sum)
           |     |      /       |
    Cin ---|     |-----/        |
           | AND |              |
    A -----|     |-----\        |
           |     |      \       |
    B -----|     |       | AND  |
           |     |      /       |
    Cin ---|     |-----/        |
           | AND |              |
    A -----|     |              |
           |     |              |
    B -----|     |              |
           |     |              |
    Cin ---|     |              |
                   | OR |--- Cout (Carry Out)

In the diagram, the XOR gates compute the sum, and the AND-OR combination computes the carry out.

Example

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

Using the truth table:

A B Cin S (Sum) Cout (Carry Out)
1 1 0 0 1

Using the formulas:

[ S = A \oplus B \oplus Cin = 1 \oplus 1 \oplus 0 = 0 ] [ Cout = (A \cdot B) + (B \cdot Cin) + (A \cdot Cin) = (1 \cdot 1) + (1 \cdot 0) + (1 \cdot 0) = 1 + 0 + 0 = 1 ]

Both methods yield the same result: S = 0 and Cout = 1.