## Half Subtractor and Full Subtractor

Table of Contents

Half Subtractor and Full Subtractor, **Half Subtractor** **Truth Table**, Full Subtractor Truth Table

**Half Subtractor (HS)**

A half subtractor is a difference and borrow combination circuit with two inputs and two outputs. The output (Borrow) shows whether a 1 has been borrowed, as well as the difference between the two binary bits input. A is referred to as a Minuend bit in the subtraction (A-B), and B is referred to as a Subtrahend bit.

**Block diagram :**

**Half Subtractor** **Truth Table**

The SOP form of the Diff and Borrow is as follows:

Diff= A’B+AB’

Borrow = A’B

**Implementation**

**Logical Expression**

Difference = A XOR B

Borrow = \overline{A}B

## Full Subtractor

A full subtractor is a combinational circuit that subtracts two bits, one of which is a minuend and the other a subtrahend, while also borrowing the lower minuend bit that was previously adjacent. Three inputs and two outputs make up this circuit. 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. Although subtraction is usually achieved by adding the complement of subtrahend to the minuend,. It is of academic interest to work out the Truth Table and logic realisation of a full subtractor; x is the minuend; y is the subtrahend; z is the input borrow; D is the difference; and B denotes the output borrow. The corresponding maps for logic functions for outputs of the full subtractor namely difference and borrow.

### Full Subtractor Truth Table (truth table of full subtractor)

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

**Full Subtractor Circuit**

**Implementation** of Full Subtractor using Half Subtractors – 2 Half Subtractors. And an OR gate is require to implement a Full Subtractor.

### What is the difference between half subtractor and full subtractor?

The difference between half subtractor and full subtractor is; the output of the half subtractor is Ex-OR of two inputs. However, the difference output of the full subtractor is Ex-OR of three inputs. The borrow output of the full subtractor is obtaine by ORing . The borrow outputs of the first and second half subtractors, respectively.