A full subtractor performs subtraction operation on two bits, a minuend and a subtrahend, and also takes into consideration whether a ‘1‘ has already been borrowed by the previous adjacent lower minuend bit or not.

**Full Subtractor**

A full subtractor performs subtraction operation
on two bits, a minuend and a subtrahend, and also takes into consideration
whether a ‘1‘ has already been borrowed by the previous
adjacent lower minuend bit or not. As a result, there are three bits to be
handled at the input of a full subtractor, namely the two bits to be subtracted
and a borrow bit designated as Bin . There are two outputs, namely the
DIFFERENCE output D and the BORROW output Bo.

The BORROW output bit tells whether the minuend
bit needs to borrow a ‘1‘ from the next possible higher minuend bit.
Figure 3.11 shows the truth table of a full subtractor.

The Boolean expressions for the two output
variables are given by the equations

The Karnaugh maps for the two expressions are
given in Fig. 3.12(a) for DIFFERENCE output D and in Fig. 3.12(b) for BORROW
output Bo. As is clear from the two Karnaugh maps, no simplification is
possible for the difference output D. The simplified expression for Bo is given
by the equation

If we compare these expressions with those
derived earlier in the case of a full adder, we find that the expression for
DIFFERENCE output D is the same as that for the SUM output. Also, the
expression for BORROW output Bo is similar to the expression for CARRY-OUT Co.
In the case of a half-subtractor, the A input is complemented. By a similar
analysis it can be shown that a full subtractor can be implemented with
half-subtractors in the same way as a full adder was constructed using
half-adders. Relevant logic diagrams are shown in Figs 3.7(a) and (b) corresponding
to Figs 3.7(a) and (b) respectively for a full adder. Again, more than one full
subtractor can be connected in cascade to perform subtraction on two larger
binary numbers. As an illustration, Fig. 3.13 shows a four-bit subtractor.

Four Bit Subtractor

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Digital Logic Circuits : Combinational Circuits : Full Subtractor |

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