This post is heavily derived from the Wikipedia post on [Digital Comparators](https://en.wikipedia.org/wiki/Digital_comparator) and therefore can be distributed under the Creative Commons Attribution-ShareAlike 3.0 license.
To compare two binary numbers, we need to find a way to check if two single bits are equal or if one is greater than the other.
## Checking Single Bit Equality
Let's construct the truth table:
| A | B | $A = B$ |
| ---- | ---- | ------- |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 1 | 1 |
The only two ways this evaluates to true is if
- A is 1 and B is 1
- A is 0 and B is 0
Therefore,
$$
(A = B) \equiv AB + \bar{A}\bar{B} \equiv \overline{(A \oplus B)}
$$
## Single Bit Comparator
To check if one bit is greater than the other, let us first look at the truth table:
| A | B | $\bar{B}$ | $A > B$ |
| ---- | ---- | --------- | ------- |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 1 | 0 | 0 |
The only way for this to evaluate to true is if A is 1 and B is 0. That is $(A > B) \equiv A\bar{B}$
## L-Bit Comparator
To compare an entire bitstring, we start from the most significant bit and check to see if one bit is greater than the other. If not, it will then check the next bit while confirming that all the previous bits were the same. For a 3-bit comparator, the logic will look like the following:
