Boolean Algebra

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

George Boole color.jpg

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:

Screen Shot 2016-05-04 at 5.49.41 PM.png

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:

 Screen Shot 2016-05-04 at 5.49.36 PM.png

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:

Screen Shot 2016-05-04 at 5.49.45 PM

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

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