Subtractor

${\displaystyle D_i=X_{}\oplus Y_i\oplus B_i}$

${\displaystyle B_{i+1}=X_i<(Y_i+B_i)}$,

where ⊕ represents exclusive or.

Subtractors are usually implemented within a binary adder for only a small cost when using the standard two's complement notation, by providing an addition/subtraction selector to the carry-in and to invert the second operand.

${\displaystyle -B=\overline{B}+1}$ (definition of two's complement notation)

${\displaystyle \begin{array}{rl}A-B& =A+(-B)\\ & =A+\overline{B}+1\end{array}}$

Half subtractor

The half subtractors can be designed through the combinational Boolean logic circuits ^[2] as shown in Figure 1 and 2. The half subtractor is a combinational circuit which is used to perform subtraction of two bits. It has two inputs, the minuend ${\displaystyle X}$ and subtrahend ${\displaystyle Y}$ and two outputs the difference ${\displaystyle D}$ and borrow out ${\displaystyle B_{\text{out}}}$. The borrow out signal is set when the subtractor needs to borrow from the next digit in a multi-digit subtraction. That is, ${\displaystyle B_{\text{out}}=1}$ when ${\displaystyle X<Y}$. Since ${\displaystyle X}$ and ${\displaystyle Y}$ are bits, ${\displaystyle B_{\text{out}}=1}$ if and only if ${\displaystyle X=0}$ and ${\displaystyle Y=1}$. An important point worth mentioning is that the half subtractor diagram aside implements ${\displaystyle X-Y}$ and not ${\displaystyle Y-X}$ since ${\displaystyle B_{\text{out}}}$ on the diagram is given by

${\displaystyle B_{\text{out}}=\overline{X}\cdot Y}$.

This is an important distinction to make since subtraction itself is not commutative, but the difference bit ${\displaystyle D}$ is calculated using an XOR gate which is commutative.

The truth table for the half subtractor is:

Using the table above and a Karnaugh map, we find the following logic equations for ${\displaystyle D}$ and ${\displaystyle B_{\text{out}}}$:

${\displaystyle D=X\oplus Y}$

${\displaystyle B_{\text{out}}=\overline{X}\cdot Y}$.

Consequently, a simplified half-subtract circuit, advantageously avoiding crossed traces in particular as well as a negate gate is:

X ── XOR ─┬─────── |X-Y|, is 0 if X equals Y, 1 otherwise ┌──┘ └──┐ Y ─┴─────── AND ── borrow, is 1 if Y > X, 0 otherwise

where lines to the right are outputs and others (from the top, bottom or left) are inputs.