# Introduction The problem asks us to **design and diagram a 3-bit subtractor** that computes the difference \( a - b \) using two 3-bit variables, \( a \) and \( b \). This is a **digital logic** design problem, common in digital electronics and computer architecture. A 3-bit subtractor takes two 3-bit binary numbers and outputs their difference, also as a binary number (possibly with a borrow bit if negative). The most common method for binary subtraction in hardware is to use **2's complement addition**. --- # Step-by-Step Solution ## Step 1: Understanding 2's Complement Subtraction Instead of directly subtracting \( b \) from \( a \), we convert subtraction into addition: \[ a - b = a + (\text{2's complement of } b) \] The **2's complement** of a number is found by: 1. Inverting all bits (1's complement) 2. Adding 1 to the result ## Step 2: Variables and Bit Representation Let: - \( a = a_2a_1a_ \) (3 bits: \( a_2 \) is MSB, \( a_ \) is LSB) - \( b = b_2b_1b_ \) (3 bits: \( b_2 \) is MSB, \( b_ \) is LSB) ## Step 3: 2's Complement Circuit for \( b \) To get the 2's complement of \( b \): - **Invert each bit of \( b \) using NOT gates:** - \( \overline{b_2} \) - \( \overline{b_1} \) - \( \overline{b_} \) - **Add 1** to the result. This is done by setting the initial carry-in of the adder to 1. ## Step 4: Using a 3-bit Adder We use a **3-bit binary adder** (chain of 3 full adders): - Inputs to the adder: - First input: \( a_2a_1a_ \) - Second input: \( \overline{b_2}\,\overline{b_1}\,\overline{b_} \) - Initial carry-in (\( C_ \)): 1 (to add the extra '1' for 2's complement) The adder outputs: - Sum bits: \( S_2, S_1, S_ \) (the result \( a-b \)) - Carry-out: (can be used to detect overflow/borrow) ## Step 5: Block Diagram Below is a conceptual diagram in text: ``` a_2 ---\ a_1 ----|------> 3-BIT ------> S_2 (MSB) a_ ---/ FULL ADDER S_1 S_ (LSB) b_2 --|NOT|--\ b_1 --|NOT|---|----/ b_ --|NOT|--/ | INVERT b | Initial Carry-In = 1 (for 2's complement) ``` ### Diagram (ASCII Art) ``` a_2 a_1 a_ | | | | | | --------------- | | | Full Adder |----> S_2, S_1, S_ (Result) | | --------------- | | | | | | ~b_2 ~b_1 ~b_ | | | NOT NOT NOT | | | b_2 b_1 b_ Carry-in (C_) = 1 ``` --- # Step-by-Step Explanation 1. **Invert all bits of \( b \)**: Use NOT gates to get \( \overline{b_2}, \overline{b_1}, \overline{b_} \). 2. **Connect \( a \) and inverted \( b \) to a 3-bit adder**: Each corresponding bit of \( a \) and \( \overline{b} \) is input to the adder. 3. **Set initial carry-in to 1**: This adds the '1' needed for 2's complement. 4. **Adder computes the result**: The adder outputs \( S_2S_1S_ \) which is \( a - b \) in binary. 5. **Borrow (if needed)**: If the result is negative, the adder's carry-out can indicate a borrow. --- # Summary Table | Step | Operation | Logic Gate/Component | |-----------------------|-------------------------------------|---------------------| | Invert b | \( \overline{b_i} \) | NOT gate | | Add a + \( \overline{b} \) + 1 | Full adder (3 bits) | Full adder | | Output | \( S_2, S_1, S_ \) (a - b) | Output lines | --- # Conclusion To **design a 3-bit subtractor**, invert the second input, add it to the first input with an initial carry-in of 1 using a 3-bit adder. This effectively computes \( a - b \) using 2's complement logic. If you need a drawn diagram (not ASCII art), let me know!