Boolean Algebra

Boolean algebra is often referred to as the algebra of logic, because the English mathematician George Boole, who is largely responsible for its beginnings, was the first to apply algebraic techniques to logical methodology. Boole showed that logical propositions and their connectives could be expressed in the language of set theory. Thus, Boolean algebra is also the algebra of sets. Algebra, in general, is the language of mathematics, together with the rules for manipulating that language. Beginning with the members of a specific set (called the universal set), together with one or more binary operations defined on that set, procedures are derived for manipulating the members of the set using the defined operations, and combinations of those operations. Both the language and the rules of manipulation vary, depending on the properties of elements in the universal set. For instance, the algebra of real numbers differs from the algebra of complex numbers, because real numbers and complex numbers are defined differently, leading to differing definitions for the binary operations of addition and multiplication, and resulting in different rules for manipulating the two types of numbers. Boolean algebra consists of the rules for manipulating the subsets of any universal set, independent of the particular properties associated with individual members of that set. It depends, instead, on the properties of sets. The universal set may be any set, including the set of real numbers or the set of complex numbers, because the elements of interest, in Boolean algebra, are not the individual members of the universal set, but all possible subsets of the universal set.