# Boolean Algebra

Today I thought I would give you a short introduction on Boolean Algebra.

George Boole

Boolean Algebra was named after the English mathematician, George Boole (1815 – 1864), who established a system of logic which is used in computers today. In Boolean algebra there are only two possible outcomes: either 1 or 0.

It must be noted that Boolean numbers are not the same as binary numbers as they represent a different system of mathematics from real numbers, whereas binary numbers is simply a alternative notation for real numbers.

### Operations

Boolean logic statements can only ever be true or false, and the words ‘AND’, ‘OR’ and ‘NOT’ are used to string these statements together.

OR can be rewritten as a kind of addition:

0 + 0 = 0 (since “false OR false” is false)
1 + 0 = 0 + 1 = 1 (since “true OR false” and “false OR true” are both true)
1 + 1 = 1 (since “true OR true” is true)

OR is denoted by:

AND can be rewritten as a kind of multiplication:

0 x 1 = 1 x 0 = 0 (since “false AND true” and “true AND false” are both false)
0 x 0 = 0 (since “false AND false” is false)
1 x 1 = 1 (since “true AND true” is true)

AND is denoted by:

NOT can be defined as the complement:

If A = 1, then NOT A = 0
If A = 0, then NOT A = 1
A + NOT A = 1 (since “true OR false” is true)
A x NOT A = 0 (since “true AND false” is false)

This is denoted by:

Venn diagrams these operations | Source: Wikipedia

Expressions in Boolean algebra can be easily simplified. For example, the variable B in A + A x B is irrelevant as, no matter what value B has, if A is true, then A OR (A AND B) is true. Hence:

A + A x B = A

Furthermore, in Boolean algebra there is a sort of reverse duality between addition and multiplication, depicted by de Morgan’s Laws:

(A + B)’ = A‘ x B‘ and (A x B)’ = A‘ + B

### Uses

In 1938, Shannon proved that Boolean algebra (using the two values 1 and 0) can describe the operation of a two-valued electrical switching circuit. Thus, in modern times, Boolean algebra is indispensable in the design of computer chips and integrated circuits.

Sources: 1 | 2 | 3