3-bit magnitude comparator

A magnitude comparator compares two binary numbers and indicates whether $A>B$, $A=B$ or $A<B$. A 3-bit comparator compares $A = A_2A_1A_0$ with $B = B_2B_1B_0$ (6 inputs, 64 combinations, 3 outputs).

Operation

Comparison starts at the most significant bit:

1. If $A_2 \ne B_2$, the result is decided: $A_2 = 1, B_2 = 0$ gives $A>B$; $A_2 = 0, B_2 = 1$ gives $A<B$.
2. If $A_2 = B_2$, compare $A_1$ and $B_1$ in the same way.
3. If $A_2 = B_2$ and $A_1 = B_1$, compare $A_0$ and $B_0$.
4. If all three pairs are equal, $A = B$.

Bit equality is detected by X-NOR gates:

$$x_i = A_iB_i + \overline{A_i}\,\overline{B_i}, \quad i = 0, 1, 2$$

Truth table (condensed)

The full table has 64 rows; it is written compactly by bit priority (X = any value):

Sample rows: $A=101, B=011$ gives $A>B$ (decided at $A_2$); $A=110, B=111$ gives $A<B$ (decided at $A_0$); $A=B=010$ gives $A=B$.

Output expressions

$$\begin{aligned} (A=B) &= x_2x_1x_0 \\ (A>B) &= A_2\overline{B_2} + x_2A_1\overline{B_1} + x_2x_1A_0\overline{B_0} \\ (A<B) &= \overline{A_2}B_2 + x_2\overline{A_1}B_1 + x_2x_1\overline{A_0}B_0 \end{aligned}$$

Circuit

A2,B2 ─[XNOR]─ x2     A1,B1 ─[XNOR]─ x1
A0,B0 ─[XNOR]─ x0

x2, x1, x0 ──────────[AND]──────────── A=B

A2, B2' ─────────[AND]──┐
x2, A1, B1' ─────[AND]──┼─[OR]──────── A>B
x2, x1, A0, B0' ─[AND]──┘

A2', B2 ─────────[AND]──┐
x2, A1', B1 ─────[AND]──┼─[OR]──────── A<B
x2, x1, A0', B0 ─[AND]──┘

Gates: 3 X-NOR, 7 AND, 2 OR (plus inverters for the complemented bits). The $A<B$ output can also be obtained as $\overline{(A>B) + (A=B)}$ using one NOR gate.