British mathematician George Boole developed a form
of algebra that uses symbolic logic to define relationships in sets  unions and intersections
to be precise. In electrical engineering, we do not usually bother ourselves with all the
rigors and formalities of the pure mathematical application. Rather, since engineers actually
build things that perform useful functions, most of us can get by with the basic operations
presented below.
Venn Diagrams do not help much when designing a digital circuit where complex (or simple
for that matter) conditional decisions must be made to determine the proper output of a multiplexer
or an entire production line controller circuit. Instead, we use Truth Tables, State Machines
and Karnaugh Maps.
There are three fundamental operators from which all other Boolean functions can be derived
 the OR gate, the AND gate, and the NOT (negation) operator. The Exclusive OR gate is included
even though it is not a fundamental function, but it is very commonly used. Venn Diagrams
for each of the functions are give just for illustration. White regions are where the functions
are True, and blue areas are False.

x + 0 = x 
x · 1 = x 

x + x = 1 
x · x = 0 

x + x = x 
x · x = x 

x + 1 = 1 
x · 0 = 0 





x + y = y + x 
x · y = y · x 
x + (y + z) = (x + y) + z 

x · (y + z) = x · y + x · z 
x + y · z = (x + y) · (x + z) 
(x + y) =
x ·
y 
(x
· y)
= x +
y 
x + x · y = x 
x · (x + y) = x 
