# Design a full subtractor circuit with three inputs x, y, $$B_{in}$$ and two outputs Diff and $$B_{out}$$· The circuit subtracts $$x-y-B_{in}$$., where $$B_{in}$$ is the input borrow, $$B_{out}$$ is the output borrow, and Diff is the difference.

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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.

Truth table:

 Input Output x y Bin D 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

From above table, we can draw the k-map for difference and borrow

D = X’Y’Bin + XY’Bin’ + X’YBin’ + XYBin

Bout = X’Bin + X’Y + YBin

Logical operation for difference

D = X’Y’Bin + X’YBin’ + XY’Bin’ + XYBin

= Bin(X’Y’ + XY) + Bin'(XY’ + X’Y)

= Bin(X   XNOR    Y) + Bin’ (X    XOR   Y)

= Bin(X   XOR    Y)’   + Bin’ (X    XOR   Y)

= (X   XOR   Y)   XOR  Bin

Logical Expression for Borrow

Bout = X’Y’Bin + X’YBin + X’Y’Bin + XYBin

Bin(XY + X’Y’) + X’Y(Bin + Bin’)

Bin(X   XOR   Y)’ + X’Y

The final circuit will be