1. Full Adder
Definition
A Full Adder is a combinational logic circuit that performs the addition of three binary bits at a time:
- Two significant bits: A and B (normal input bits)
- One carry bit: Cin (carry from a previous lower significant stage)
It produces:
- Sum (S) — result bit
- Carry (Cout) — output carry bit to the next stage
Inputs and Outputs
- Inputs: A, B, Cin
- Outputs: Sum (S), Carry Out (Cout)
Truth Table
| A | B | Cin | Sum (S) | Carry (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 |
Boolean Expressions
- Sum (S) = (A ⊕ B) ⊕ Cin
- Carry (Cout) = (A · B) + (B · Cin) + (A · Cin)
Logic Diagram (Simple Description)
The Full Adder can be built using:
- Two Half Adders + One OR gate
Structure:
A ----|⊕|----\
| \
B ----|⊕| ⊕---- Sum (S)
| /
Cin ---|⊕|----/
(A·B) --\
OR ---- Carry (Cout)
(B·Cin) -/
(A·Cin)-/
- First Half Adder: adds A and B
- Second Half Adder: adds output of first HA and Cin
- OR Gate: combines the carries from both Half Adders
2. Full Subtractor
Definition
A Full Subtractor is a combinational logic circuit that performs subtraction of three binary bits:
- Two significant bits: A and B
- A borrow bit: Bin (borrow from previous lower significant stage)
It produces:
- Difference (D) — result bit
- Borrow Out (Bout) — borrow to next stage
Inputs and Outputs
- Inputs: A, B, Bin
- Outputs: Difference (D), Borrow Out (Bout)
Truth Table
| A | B | Bin | Difference (D) | Borrow (Bout) |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 | 1 |
Boolean Expressions
- Difference (D) = (A ⊕ B) ⊕ Bin
- Borrow (Bout) = (¬A · B) + (¬(A ⊕ B) · Bin)
Where:
¬A= NOT A⊕= XOR
Logic Diagram (Simple Description)
The Full Subtractor can be built using:
- Two Half Subtractors + One OR gate
Structure:
A ----|⊕|----\
| \
B ----|⊕| ⊕---- Difference (D)
| /
Bin ---|⊕|----/
(¬A·B) --\
OR ---- Borrow (Bout)
(¬(A⊕B)·Bin) -/
- First Half Subtractor: subtracts B from A
- Second Half Subtractor: subtracts Bin from the first difference
- OR Gate: combines the two borrows
📋 Summary Table: Full Adder vs Full Subtractor
| Feature | Full Adder | Full Subtractor |
|---|---|---|
| Purpose | Adds three bits (A, B, Cin) | Subtracts three bits (A, B, Bin) |
| Inputs | A, B, Carry-in (Cin) | A, B, Borrow-in (Bin) |
| Outputs | Sum, Carry-out (Cout) | Difference, Borrow-out (Bout) |
| Sum/Diff | (A⊕B)⊕Cin | (A⊕B)⊕Bin |
| Carry/Borrow | (A·B)+(B·Cin)+(A·Cin) | (¬A·B)+(¬(A⊕B)·Bin) |
| Built Using | Two Half Adders + OR gate | Two Half Subtractors + OR gate |
🌟 Key Points to Remember
- Full Adder adds three inputs and generates two outputs (Sum and Carry).
- Full Subtractor subtracts three inputs and generates two outputs (Difference and Borrow).
- XOR gate plays a major role in both circuits.
- Full circuits are necessary for multi-bit addition/subtraction (like adding two 4-bit numbers).
- Used heavily in Arithmetic Logic Units (ALU) of CPUs.
🎯 Real Life Application
- Calculators
- CPUs for addition/subtraction
- Digital counters
- ALUs in Microprocessors and Microcontrollers
- Signal processing
✏️ Final Tip for Exams
👉 Always draw the truth table and mention the Boolean expressions — they fetch full marks!
👉 Logic diagrams can be drawn neatly using XOR, AND, OR, and NOT gates.