A half adder adds two one-bit inputs, A and B. It outputs Sum = A XOR B and Carry = A AND B. A full adder adds A, B, and carry-in Cin. It outputs Sum = A XOR B XOR Cin and Cout = (A AND B) OR (Cin AND (A XOR B)). The key difference is carry-in. A half adder has no Cin input, while a full adder can accept a carry from the previous bit position.
| Feature | Half adder | Full adder |
|---|---|---|
| Inputs | Two: A and B | Three: A, B, and carry-in Cin |
| Outputs | Sum and Carry | Sum and carry-out Cout |
| Truth-table rows | Four | Eight |
| Sum | A XOR B | A XOR B XOR Cin |
| Carry output | A AND B | (A AND B) OR (Cin AND (A XOR B)) |
| Incoming carry | Cannot accept one | Accepts Cin from a previous stage |
| One gate-level form | One XOR gate and one AND gate | Two XOR gates, two AND gates, and one OR gate |
| Typical role | Adds one bit pair when no carry-in is required | Forms a one-bit stage in a wider binary adder |
Both circuits produce a one-bit sum and a carry output. A full adder is not by itself a multibit adder. Multiple full-adder stages can be connected to add wider binary words.
Half Adder Truth Table
A half adder has two inputs, so its truth table has four rows. The Sum output is A XOR B. Carry is A AND B.
| A | B | Sum | Carry |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
Source check: the XOR rows match the onsemi MM74HC86 data sheet, Rev. 3. The carry rows match the onsemi 74VHC08 data sheet, Rev. 4.
Full Adder Truth Table
A full adder has three inputs, so its truth table has eight rows. Sum is A XOR B XOR Cin. Carry-out is (A AND B) OR (Cin AND (A XOR B)).
| 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 |
All eight rows match the one-stage truth table in the onsemi MC14008B data sheet, Rev. 9.
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What Is a Half Adder?
A half adder is a two-input combinational logic circuit. Its output depends only on the current values of A and B. The XOR output becomes high when exactly one input is high. The AND output becomes high only when both inputs are high.
This makes the half adder suitable when no incoming carry must be included. It can handle the least significant bit of an addition when the operation has no external carry-in. It cannot process a carry generated by an earlier bit position.
What Is a Full Adder?
A full adder is a three-input combinational logic circuit. It adds A, B, and Cin, then produces Sum and Cout. Sum is high when an odd number of the three inputs are high. Cout is high when at least two inputs are high.
One full adder handles one bit position. Its carry output can feed the carry input of the next position. The MC14008B data sheet shows four full-adder stages in one device and also shows devices connected for a 16-bit addition.
How to Build a Full Adder Using Half Adders
A complete full adder can be built from two half adders and one OR gate. Two half adders alone are not enough because their separate carry terms still need to be combined.
- The first half adder receives A and B. It produces X = A XOR B and C1 = A AND B.
- The second half adder receives X and Cin. It produces Sum = X XOR Cin and C2 = X AND Cin.
- The OR gate combines the carries: Cout = C1 OR C2.
Substituting X gives Sum = A XOR B XOR Cin and Cout = (A AND B) OR (Cin AND (A XOR B)). These expressions produce every row in the full-adder truth table above.
When transferring these logic relationships into a drawing, use consistent symbols and net labels. OurPCB's schematic guide explains how logic diagrams represent circuit relationships. If the design uses packaged logic, the PCB components guide provides a practical overview of IC identification and placement.
Half Adder and Full Adder Applications
Use a half adder when two bits must be added without an incoming carry. In a multibit addition with no external carry-in, a half adder can process the least significant bit.
Use full-adder stages for the remaining bit positions because each stage can accept the carry from the preceding position. Connecting Cout from one stage to Cin of the next creates a wider binary adder. A full-adder IC can also use internal carry logic, as the MC14008B does across four stages.
Summary
A half adder has two inputs and no carry-in. Its outputs are Sum = A XOR B and Carry = A AND B. A full adder adds the Cin input, uses an eight-row truth table, and produces Cout = (A AND B) OR (Cin AND (A XOR B)).
To construct a full adder from half adders, use two half adders and an OR gate. The OR gate combines the two carry terms into Cout.
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