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.
A set is a collection of objects, called members or elements. The members of a set can be physical objects, such as people, stars, or red roses, or they can be abstract objects, such as ideas, numbers, or even other sets. A set is referred to as the universal set (usually called I) if it contains all the elements under consideration. A set, S, not equal to I, is called a proper subset of I, if every element of S is contained in I. This is written and read "S is contained in I."
If S equals I, then S is called an improper subset of I, that is, I is an improper subset of itself (note that two sets are equal if and only if they both contain exactly the same elements). The special symbol is given to the set with no elements, called the empty set or null set. The null set is a subset of every set.
When dealing with sets there are three important operations. Two of these operations are binary (that is, they involve combining sets two at a time), and the third involves only one set at a time. The two binary operations are union and intersection. The third operation is complementation. The union of two sets S and T is the collection of those members that belong to either S or T or both.
The intersection of the sets S and T is the collection of those members that belong to both S and T, and is written
The complement of a subset, S, is that part of I not contained in S, and is written S'.
The properties of Boolean algebra can be summarized in four basic rules.
(1) Both binary operations have the property of commutativity, that is, order doesn't matter.
S T = T S, and S T = T S.
(2) Each binary operation has an identity element associated with it. The universal set is the identity element for the operation of intersection, and the null set is the identity element for the operation of union.
S I = S, and S ø = S.
(3) each operation is distributive over the other.
S (T V) = (S T) (S V), and S (T V) = (S T) U (S V).
This differs from the algebra of real numbers, for which multiplication is distributive over addition, a(b+c) = ab + ac, but addition is not distributive over multiplication, a+(bc) not equal (a+b)(a+c).
(4) each element has associated with it a second element, such that the union or intersection of the two results in the identity element of the other operation.
A A' = I, and A A' = ø.
This also differs from the algebra of real numbers. Each real number has two others associated with it, such that its sum with one of them is the identity element for addition, and its product with the other is the identity element for multiplication. That is, a + (-a) = 0, and a(1/a) = 1.
The usefulness of Boolean algebra comes from the fact that its rules can be shown to apply to logical statements. A logical statement, or proposition, can either be true or false, just as an equation with real numbers can be true or false depending on the value of the variable. In Boolean algebra, however, variables do not represent the values that make a statement true, instead they represent the truth or falsity of the statement. That is, a Boolean variable can only have one of two values. In the context of symbolic logic these values are true and false. Boolean algebra is also extremely useful in the field of electrical engineering. In particular, by taking the variables to represent values of on and off (or 0 and 1), Boolean algebra is used to design and analyze digital switching circuitry, such as that found in personal computers, pocket calculators, cd players, cellular telephones, and a host of other electronic products.
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