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 