By Muhammad Ibrahim · Published June 18, 2026 · Updated October 29, 2026 · 8 min read
How to Build a Half Adder and Full Adder with 74-Series Gates
A half adder adds two bits. A full adder adds two bits plus a carry-in, which is what lets adders chain into multi-bit arithmetic. Both are built from the same three gate types, and both are worth deriving from the truth table rather than copying from a diagram. This page does the derivation for both, then builds them from real 74-series parts. The circuit you can open below is the half adder from my Lab 2, and the full adder has its own tutorial with a circuit file.
- HALF ADDER
- 1 XOR + 1 AND
- FULL ADDER
- 2 XOR + 2 AND + 1 OR
- CHIPS
- 7486, 7408, 7432
- CIRCUIT FILE
- Lab 2 half adder
Components needed for the half adder and full adder
- One 7486 (quad XOR), one 7408 (quad AND) and one 7432 (quad OR)
- A DIP switch. Two switches are used for the half adder and three (A, B, Cin) for the full adder
- A Light Bar. Two LEDs are used, one for SUM and one for the carry
- A Regulated +5V Power part, a Ground (GND) part and one half-size breadboard
- The Lab 2 circuit file above also uses a 7404 (hex inverter), because it has no XOR chip
Open the circuit
Open the half adder from Lab 2. It makes SUM from inverters, AND gates and an OR gate instead of an XOR chip, so it has the same truth table as the XOR version described below. Switch to Simulate and flip A and B. The SUM and CAR LEDs show the result.
How to build a half adder and full adder step by step
Write the half adder truth table
For inputs A and B the outputs are SUM and CARRY. A=0, B=0 gives SUM 0 and CARRY 0. A=0, B=1 and A=1, B=0 give SUM 1 and CARRY 0. A=1, B=1 gives SUM 0 and CARRY 1. SUM is 1 when exactly one input is 1, which is XOR. CARRY is 1 only when both are 1, which is AND.
Choose how to build SUM
With a 7486 the sum is a single XOR gate. Without one, write the XOR as AB' + A'B and use two inverters, two AND gates and an OR gate. The circuit file above is the second version, and the Lab 2 tutorial explains it pin by pin. Both give the same truth table.
Power the ICs correctly
The 7486, 7408 and 7432 are all 14-pin quad two-input gate packages with the same pin layout: pin 14 is V+ and pin 7 is ground. Each gate is two inputs followed by an output (pins 1 and 2 give 3, pins 4 and 5 give 6, pins 10 and 9 give 8, pins 13 and 12 give 11). Connect V+ and ground on every chip before wiring any logic. An unpowered TTL chip is the most common cause of a circuit that does nothing.
Wire the half adder with an XOR chip
Connect A to pin 1 and B to pin 2 of the 7486, and take SUM from pin 3. Connect A to pin 1 and B to pin 2 of the 7408, and take CARRY from pin 3. Connect an LED to each output, switch to Simulate and step through all four combinations to confirm the table.
Extend it to a full adder
A full adder is two half adders and an OR gate. Half adder 1 takes A and B and produces S1 and C1. Half adder 2 takes S1 and Cin and produces the final SUM and C2. Cout = C1 OR C2. That is two XOR gates, two AND gates and one OR gate in total. The full adder tutorial linked below wires every pin and has its own circuit file.
Verify all eight input combinations
Step A, B and Cin through 000 to 111. The SUM and carry-out outputs read together as a binary number must equal the number of 1s among the three inputs. For example 1+1+0 = 10 (SUM 0, Cout 1) and 1+1+1 = 11 (SUM 1, Cout 1).
Half adder truth table
| A | B | SUM | CARRY |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
Full adder truth table
| 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 |
In every row, A + B + Cin = 2 × Cout + SUM.
Adder gate counts
| Adder | Gates | Chips |
|---|---|---|
| Half adder with an XOR chip | 1 XOR, 1 AND | 7486, 7408 |
| Half adder without an XOR chip (the circuit file above) | 2 NOT, 3 AND, 1 OR | 7404, 7408, 7432 |
| Full adder | 2 XOR, 2 AND, 1 OR | 7486, 7408, 7432 |
How the adders work: equations, gate count and ripple carry
The full adder equations
SUM = A ⊕ B ⊕ Cin and Cout = A·B + Cin·(A ⊕ B). The second form is exactly the two-half-adder construction: A·B is the first half adder's carry and Cin·(A ⊕ B) is the second half adder's carry. Because the two carries can never both be 1 at the same time, an OR gate combines them without losing information.
Gate count and the 74-series packages
One full adder needs 2 XOR + 2 AND + 1 OR = 5 gates, which fits in one 7486 (using 2 of its 4 gates), one 7408 (2 of 4) and one 7432 (1 of 4). A 4-bit ripple-carry adder repeats this four times, which is why the 7483 4-bit adder exists as a single package.
Ripple-carry delay
In a chain of full adders each stage must wait for the carry from the previous one. Delay therefore grows linearly with word length, which is the reason faster adders (carry-lookahead) are used in practice. In a simulation you can see this by toggling the lowest input of a multi-bit adder and watching the carry settle.
Expected half adder and full adder output
- Half adder: SUM = A XOR B and CARRY = A AND B for all four input pairs.
- Full adder: SUM is 1 for an odd number of 1s among A, B and Cin, and Cout is 1 when at least two inputs are 1.
- Rows 011, 101, 110 and 111 all produce Cout = 1.
Adder not working? Common mistakes
Outputs never change
Check pin 14 (V+) and pin 7 (ground) on all three ICs. Also confirm you are using the output pin of each gate (3, 6, 8 or 11), not an input pin.
Carry-out is wrong only for 1+1+1
That row needs both C1 and C2, so check that the OR gate receives both carry signals, and that half adder 2 uses S1 (not A) as its first input.
Unused gate inputs behave unpredictably
Tie the inputs of any gate you are not using to a defined level. Floating TTL inputs tend to read high but are noise-sensitive on real hardware, so it is a good habit to practise in simulation.
Frequently asked questions
What is the difference between a half adder and a full adder?
A half adder adds two bits and has no carry input, so it can only be the first stage of an adder. A full adder also takes a carry-in, so several of them can be chained to add numbers of any length.
How many gates does a full adder need?
Five: two XOR gates, two AND gates and one OR gate. They fit on one 7486 (2 of its 4 gates), one 7408 (2 of 4) and one 7432 (1 of 4).
Can a half adder be built without an XOR gate?
Yes. Write the XOR as AB' + A'B and build it from two inverters, two AND gates and an OR gate. The half adder circuit on this page does exactly that with a 7404, a 7408 and a 7432, and the Lab 2 tutorial wires it pin by pin.
How do you build a 4-bit adder from full adders?
Use four full adders and connect the Cout of each one to the Cin of the next, with the first Cin tied to 0. This is a ripple-carry adder. The 7483 packs the same four stages into a single chip.
Try it yourself
Open LogicBench and follow along — the simulator is free and needs no account.
Launch the simulator