Full Adder Calculator
Calculate the Sum and Carry-Out of a full adder using binary inputs A, B and Carry-In. Inspect XOR, AND and OR logic, Boolean equations, truth-table values and step-by-step binary addition.
What Is a Full Adder?
A full adder is a combinational digital circuit that adds three one-bit binary inputs. Two inputs represent the binary operand bits A and B, while the third input is the incoming carry, normally written as Cin.
The circuit produces two outputs: a Sum bit and a Carry-Out bit. Carry-Out can then become the Carry-In of the next more significant adder stage.
This ability to process an incoming carry makes the full adder suitable for constructing multi-bit binary adders.
Full Adder Boolean Equations
Sum Equation
The Sum bit is obtained by XORing A and B and then XORing that result with Carry-In.
Carry-Out Equation
Carry-Out becomes 1 whenever at least two of the three full-adder inputs are 1.
Alternative Carry-Out Expression
The full-adder carry can also be expressed using three pairwise AND terms. This form makes the majority behavior of the carry output easy to see.
If two or all three inputs are 1, Carry-Out is 1.
Full Adder Truth Table
| A | B | Carry-In | Sum | Carry-Out | Binary Result | Decimal Total |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 00 | 0 |
| 0 | 0 | 1 | 1 | 0 | 01 | 1 |
| 0 | 1 | 0 | 1 | 0 | 01 | 1 |
| 0 | 1 | 1 | 0 | 1 | 10 | 2 |
| 1 | 0 | 0 | 1 | 0 | 01 | 1 |
| 1 | 0 | 1 | 0 | 1 | 10 | 2 |
| 1 | 1 | 0 | 0 | 1 | 10 | 2 |
| 1 | 1 | 1 | 1 | 1 | 11 | 3 |
Worked Example: A = 1, B = 0, Cin = 1
B = 0
Cin = 1
A XOR B
= 1 XOR 0
= 1
Sum
= 1 XOR Cin
= 1 XOR 1
= 0
Carry-Out
= (1 AND 0) OR (1 AND 1)
= 0 OR 1
= 1
Output = 10
Decimal total = 2
Worked Example: A = 1, B = 1, Cin = 1
B = 1
Cin = 1
1 + 1 + 1 = 3 decimal
3 decimal = 11 binary
Sum = 1
Carry-Out = 1
Final output = 11
How Full Adder Logic Works
First XOR Operation
The first XOR operation combines A and B. Its output is then combined with Cin to determine the final Sum bit.
Second XOR Operation
The intermediate XOR result and Carry-In determine the Sum output.
Carry Generate
When A and B are both 1, the stage directly generates a carry.
Carry Through Cin
Carry-In can also create Carry-Out when the XOR result of A and B is 1.
Full Adder vs Half Adder
| Feature | Half Adder | Full Adder |
|---|---|---|
| Binary operand inputs | A, B | A, B |
| Carry-In | No | Yes |
| Total inputs | 2 | 3 |
| Sum output | Yes | Yes |
| Carry output | Yes | Yes |
| Previous-stage carry | Cannot accept | Can accept |
| Multi-bit adder stages | Limited | Suitable |
Full Adder Using Two Half Adders
A full adder can conceptually be constructed from two half adders plus an OR gate.
A + B
→ intermediate Sum S1
→ Carry C1
Half Adder 2:
S1 + Cin
→ final Sum
→ Carry C2
Final Carry-Out:
Cout = C1 OR C2
This decomposition helps show the relationship between half-adder and full-adder logic.
Full Adders in Multi-Bit Binary Addition
For wider binary numbers, multiple full-adder stages can be connected together. The Carry-Out of a lower-order stage becomes the Carry-In of the next higher-order stage.
FA1: A1 + B1 + C1 → S1 + C2
FA2: A2 + B2 + C2 → S2 + C3
FA3: A3 + B3 + C3 → S3 + C4
This stage-to-stage connection is the basic principle behind a ripple carry adder.
Important Full Adder Notes
The output consists of two bits: Carry-Out and Sum.
The Sum equation is A XOR B XOR Cin.
Carry-Out becomes 1 whenever at least two of the three inputs are 1.
For A = 1, B = 1 and Cin = 1, the total is decimal 3, so the full-adder output is binary 11.