Circuit Design
Ultimately the point of boolean algebra is to facilitate good circuit design and ultimately to produce electronic equipment. Simplification and conversion to functionally complete sets are just a means to this end. All of the operators we have encountered can exist as logic gates in a circuit, and by combining these gates we can produce useful circuits - or "components".
Simplification of Boolean Algebra (If Only...)
Some of these algebraic expressions are starting to get quite complicated, but fear not there are tools we can use to simplify them by the boolean equivalent of factorisation.
More Boolean Operators
Using the three primary boolean operators (AND, NOT and OR) it is possible to derive a number of other operators, these are not unique - in the sense that they can be replicated by combining one or more of the primary operators in order to form them. Examples of these derived operators are XOR, NAND and NOR. Any set of operators which can be used to perform any desired operation is known as a functionally complete set, AND/OR/NOT is a functionally complete set since any derived operator can be formed from the base set. NAND and NOR are also individually functionally complete sets as either of them can be used to perform any operation.
De Morgan's Theorum
Augustus De Morgan was a 19th century mathematician working in the field of propositional logic, he established a series of rules concerned with conversion between different operators using inversion. The fundamental basis of his theory is that NOT ( P AND Q) = (NOT P) OR (NOT Q), and similarly NOT (P OR Q) = (NOT P) AND (NOT Q), this is often known as De Morgan Duality.
In The Beginning... There was 1 and 0
Boolean Algebra is a two state system of algebra, we can apply this to the binary number system where only two valid numeric symbols exist, "1" and "0" and thus use boolean logic to solve binary problems. The electronics inside a computer rely on this system, modeling voltage in a connection as either on (1, or TRUE) or off (0, or FALSE).