Boolean algebra is algebra with only two values, 0 and 1, and three basic operations: AND, OR and NOT. It is named after George Boole, who used it in the 1800s to describe logical reasoning. In the 1930s it was applied to switching circuits, and it has been the language of digital design ever since.
Why it matters: every digital circuit computes a boolean function. Writing that function as a boolean expression lets you reason about the circuit on paper, and rewriting the expression with the laws of boolean algebra gives a smaller circuit that does the same job.
It looks like school algebra, but it is not the same:
- + means OR, so = 1, not 2.
- There is no subtraction or division, so you cannot cancel: = does not mean B = C.
- Repeating changes nothing: = A and = A.
- OR distributes over AND: = , which fails for ordinary numbers.
Because there are only two values, any claim can be checked by trying every input in a truth table.
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 | 0 |