Why Boole Chose to Endow His Operators with These Functions: Exploring Boolean Algebra

George Boole, a pioneering mathematician, was deeply involved in the creation of Boolean algebra. The operators he defined play a crucial role in modern logic and computer science. This article delves into why Boole's operators, particularly AND, OR, and NOT, were endowed with specific functions, focusing on the choice of multiplication representing intersection and the reasoning behind it.

Boolean Algebra and Its Operators

Boolean algebra, initially developed by George Boole in the mid-19th century, forms the backbone of digital logic and computer programming. Boole's system was designed to align set theory with algebraic operations, making it easier to manipulate logical expressions using mathematical notation.

Why Set Theory Aligns with Algebra

In Boolean algebra, the operators AND, OR, and NOT are represented as , ·, and 1's and 0's respectively. If we identify True with 1 and False with 0, then anything and False equals False corresponds to anything times zero equals zero. This correspondence shows the inherent connection between Boolean algebra and set theory.

Conjunction and Multiplication

When conjunction (AND) is represented by multiplication (·), it becomes clear that a and b is equivalent to a · b. This representation aligns with the concept of intersection, where the intersection of sets A and B is the subset of elements that are common to both sets. Mathematically, this can be expressed as A ∩ B, which corresponds to the product of the elements when using Boolean algebra.

Disjunction and Bounded Addition

In contrast to conjunction, disjunction (OR) does not correspond to simple addition but to a specific operation called bounded addition. The disjunction of A or B is represented as the minimum between ab and 1. This is because in Boolean algebra, the OR operation is defined such that a or b 1 if either a or b (or both) are 1, and 0 otherwise. This can be expressed as min(a b, 1).

Set Representation and Vector Spaces

Consider a set with n elements, ordered from 1 to n. Each subset can be represented by a vector of length n, where the ith element is 1 if i is in the set and 0 otherwise. These vectors naturally form a vector space of dimension n over the integers modulo 2 (Z2).

In this vector space, the elementwise sum represents the symmetric difference of sets. The Hadamard product, or elementwise product, represents the intersection. Both operations are commutative and associative. Additionally, for any polynomial in this vector space, every power of any variable is reduced to its first power.

De Morgan's Laws and Algebraic Formulation

De Morgan's laws can be simply formulated in terms of algebra. For example, 1 u 1 v 1 u v u v. This allows for the simplification of complex expressions involving intersections, unions, and complements. By writing out the formula in algebraic notation, one can easily simplify and evaluate these expressions.

Choice of Symbols and Their Functions

The choice of symbols for Boolean operations is not arbitrary. Many of the algebraic rules that arise from Boolean logic mirror the rules of arithmetic, such as A B B A and A · B B · A. The symbols for addition (AND) and multiplication (OR) were thus an obvious choice.

However, the choice of whether AND or OR should be represented by or . is intriguing. In Boolean logic, the distributive law works equally in both directions, but this is not true in arithmetic. Thus, the current choice of for AND and . for OR might make the OR form of distribution less intuitive for people who are more accustomed to the AND form.

The Convenience of Boolean Expressions

The decision to make AND be addition ( ) and OR be multiplication (·) aligns with the preferential way we write polynomials as a sum of products in regular algebra. A Boolean expression can often be written as "an ORing of ANDs," mirroring the algebraic form. This form is generally more convenient and helps in simplifying and evaluating complex Boolean expressions.

Conclusion

George Boole's choice of operators in Boolean algebra was rooted in aligning set theory with algebraic operations. The functions of AND, OR, and NOT, particularly the choice of multiplication to represent intersection, were designed to simplify logical expressions and make them more manageable. Understanding the underlying principles and historical context can enhance our appreciation of this foundational system in modern mathematics and computer science.

Keywords

Boolean algebra, operators, Boolean logic