# Full Subtractor in Digital Logic

A full subtractor is a **combinational circuit** that performs subtraction of two bits, one is minuend and other is subtrahend, taking into account borrow of the previous adjacent lower minuend bit. This circuit** has three inputs and two outputs**. The three inputs A, B and Bin, denote the minuend, subtrahend, and previous borrow, respectively. The two outputs, D and Bout represent the difference and output borrow, respectively.

**Truth Table –**

From above table we can draw the K-Map as shown for “difference” and “borrow”.

**Logical expression for difference –**

D = A’B’Bin + A’BBin’ + AB’Bin’ + ABBin = Bin(A’B’ + AB) + Bin’(AB’ + A’B) = Bin( A XNOR B) + Bin’(A XOR B) = Bin (A XOR B)’ + Bin’(A XOR B) = Bin XOR (A XOR B) = (A XOR B) XOR Bin

**Logical expression for borrow –**

Bout = A’B’Bin + A’BBin’ + A’BBin + ABBin = A’B’Bin +A’BBin’ + A’BBin + A’BBin + A’BBin + ABBin = A’Bin(B + B’) + A’B(Bin + Bin’) + BBin(A + A’) = A’Bin + A’B + BBin OR Bout = A’B’Bin + A’BBin’ + A’BBin + ABBin = Bin(AB + A’B’) + A’B(Bin + Bin’) = Bin( A XNOR B) + A’B = Bin (A XOR B)’ + A’B

**Logic Circuit for Full Subtractor –**

**Implementation** of Full Subtractor using Half Subtractors –

2 Half Subtractors and an OR gate is required to implement a Full Subtractor.

**Reference –** Full Subtractor – Wikipedia

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