Implement a full adder circuit with a decoder and two OR gates.

To implement a full adder circuit using a decoder and two OR gates, we need to understand the functionality of a full adder and how a decoder can be used to generate the necessary signals for summation and carry-out.

Full Adder Functionality

A full adder is a digital circuit that computes the addition of three binary bits: two significant bits and a carry-in bit. It outputs a sum bit and a carry-out bit. The truth table for a full adder is as follows:

A (Input)	B (Input)	Cin (Carry-in)	Sum (Output)	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

Decoder Functionality

A decoder is a combinational circuit that converts binary information from n input lines to a maximum of 2^n unique output lines. For our purpose, we will use a 3-to-8 line decoder since we have three inputs (A, B, Cin) and we need to generate signals for each possible combination of these inputs.

The truth table for a 3-to-8 line decoder is as follows:

A (Input)	B (Input)	Cin (Input)	D0	D1	D2	D3	D4	D5	D6	D7
0	0	0	1	0	0	0	0	0	0	0
0	0	1	0	1	0	0	0	0	0	0
0	1	0	0	0	1	0	0	0	0	0
0	1	1	0	0	0	1	0	0	0	0
1	0	0	0	0	0	0	1	0	0	0
1	0	1	0	0	0	0	0	1	0	0
1	1	0	0	0	0	0	0	0	1	0
1	1	1	0	0	0	0	0	0	0	1

Implementing Full Adder with Decoder and OR Gates

To implement the full adder, we need to generate the Sum and Cout signals using the outputs of the decoder (D0 to D7) and two OR gates.

Step 1: Generate Sum Signal

The Sum output is 1 for the following input combinations: 001, 010, 100, 111. These correspond to decoder outputs D1, D2, D4, and D7. We can use an OR gate to combine these signals:

Sum = D1 OR D2 OR D4 OR D7

Step 2: Generate Cout Signal

The Cout output is 1 for the following input combinations: 011, 101, 110, 111. These correspond to decoder outputs D3, D5, D6, and D7. We can use another OR gate to combine these signals:

Cout = D3 OR D5 OR D6 OR D7

Step 3: Connect Decoder Outputs to OR Gates

Now, we connect the appropriate outputs from the decoder to the inputs of the two OR gates as per the logic derived above.

Step 4: Final Circuit

The final circuit will look like this:

Inputs: A, B, Cin
Decoder: 3-to-8 line decoder
OR Gates: Two 4-input OR gates

Decoder Outputs -> OR Gate Inputs:
D1, D2, D4, D7 -> OR Gate 1 (for Sum)
D3, D5, D6, D7 -> OR Gate 2 (for Cout)

Outputs: Sum (from OR Gate 1), Cout (from OR Gate 2)

Example

Let's consider the input combination A=1, B=1, and Cin=0. According to the full adder truth table, the expected output is Sum=0 and Cout=1.

• The decoder will activate output D6 because the input is 110.
• For the Sum signal, OR Gate 1 will have inputs D1=0, D2=0, D4=0, D7=0, so the output will be 0.
• For the Cout signal, OR Gate 2 will have inputs D3=0, D5=0, D6=1, D7=0, so the output will be 1.

Thus, the full adder implemented with a decoder and two OR gates correctly computes the Sum and Cout for the given inputs.