One of the most important Boolean algebras is that with the structure 〈{0,1};+,⋅,¯,0,1〉, that is, it consists of the power set {0,1}, has two natural binary operators (+ meaning "maximum" and ⋅ meaning "minimum"), a natural unary operator (¯ meaning "complement"; due to lack of HTML symbols I will use a single quote to denote complement), the number 0 denoting a null set or ∅, and 1 denoting the full power set. This is important because of its applications in computer science, electronics, and logic, among others. This Boolean algebra and its binary operators can best be described pictorially by the use of electrical circuits.
____X____
| |
| |
| |
| |
--|----o---
Damn my infernal ASCII picture skills. But anyway. The point is, | down the bottom stands for a battery or power source; o stands for a lightbulb; X stands for a switch. When X is closed, the light will come on, in other words, the circuit is fully closed. We'll call that "1". When X is open, the light will turn off. We'll call that "0". Now what happens if we add a second switch Y to the top?
__X___Y__
| |
| |
| |
| |
--|----o---
It stands to reason that both switches must be closed for the light to come on.
| switch X |
switch Y |
circuit |
| open |
open |
open |
| open |
closed |
open |
| closed |
open |
open |
| closed |
closed |
closed |
This happens to agree exactly with the Boolean operator ⋅, or "minimum", as follows:
| x |
y |
min{x,y} |
| 0 |
0 |
0 |
| 0 |
1 |
0 |
| 1 |
0 |
0 |
| 1 |
1 |
1 |
In other words, x⋅y is the minimum value that x and y have. Say x had a value of 5, and y had a value of 7 (neither of which actually occur in Boolean algebras that I know of), then x⋅y = 5. This operation is a Boolean Multiplication. Boolean Addition, on the other hand, is finding the maximum value of x and y, as shown by two switches placed in a circuit parallel to each other.
____X____
| |
|____Y____|
| |
| |
| |
| |
--|----o---
Logically, only one of these switches needs to be on for the circuit to work. Ergo, we can change our table from before slightly:
| switch X |
switch Y |
circuit |
| open |
open |
open |
| open |
closed |
closed |
| closed |
open |
closed |
| closed |
closed |
closed |
Boolean Addition is counterintuitive to regular addition: in Boolean land, 1+1=1.
| x |
y |
max{x,y} |
| 0 |
0 |
0 |
| 0 |
1 |
1 |
| 1 |
0 |
1 |
| 1 |
1 |
1 |
These tables are called input-output tables or, as I prefer to call them, truth tables. So why is it so important? Computers work in binary digits, in other words, 0s and 1s. Logic circuits work with "and", "or" and "not" gates (which parallel ⋅, + and ', respectively). Even digital watches run using Boolean algebra of sorts: each little liquid crystal "bar" runs off of a similar Boolean algebra.