Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted by 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction denoted as, disjunction denoted as, and negation denoted as. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction, and division. Boolean algebra is therefore a formal way of describing logical operations in the same way that elementary algebra describes numerical operations.
Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic, and set forth more fully in his An Investigation of the Laws of Thought. According to Huntington, the term Boolean algebra was first suggested by Henry M. Sheffer in 1913, although Charles Sanders Peirce gave the title "A Boolian Algebra with One Constant" to the first chapter of his "The Simplest Mathematics" in 1880. Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages. It is also used in set theory and statistics.
History
A precursor of Boolean algebra was Gottfried Wilhelm Leibniz's algebra of concepts. The usage of binary in relation to the I Ching was central to Leibniz's characteristica universalis. It eventually created the foundations of algebra of concepts. Leibniz's algebra of concepts is deductively equivalent to the Boolean algebra of sets.Boole's algebra predated the modern developments in abstract algebra and mathematical logic; it is however seen as connected to the origins of both fields. In an abstract setting, Boolean algebra was perfected in the late 19th century by Jevons, Schröder, Huntington and others, until it reached the modern conception of an mathematical structure. For example, the empirical observation that one can manipulate expressions in the algebra of sets, by translating them into expressions in Boole's algebra, is explained in modern terms by saying that the algebra of sets is a Boolean algebra. In fact, M. H. Stone proved in 1936 that every Boolean algebra is isomorphic to a field of sets.
In the 1930s, while studying switching circuits, Claude Shannon observed that one could also apply the rules of Boole's algebra in this setting, and he introduced switching algebra as a way to analyze and design circuits by algebraic means in terms of logic gates. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the two-element Boolean algebra. In modern circuit engineering settings, there is little need to consider other Boolean algebras, thus "switching algebra" and "Boolean algebra" are often used interchangeably.
Efficient implementation of Boolean functions is a fundamental problem in the design of combinational logic circuits. Modern electronic design automation tools for very-large-scale integration circuits often rely on an efficient representation of Boolean functions known as binary decision diagrams for logic synthesis and formal verification.
Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed in this way. Boolean algebra is not sufficient to capture logic formulas using quantifiers, like those from first-order logic.
Although the development of mathematical logic did not follow Boole's program, the connection between his algebra and logic was later put on firm ground in the setting of algebraic logic, which also studies the algebraic systems of many other logics. The problem of determining whether the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to true is called the Boolean satisfiability problem, and is of importance to theoretical computer science, being the first problem shown to be NP-complete. The closely related model of computation known as a Boolean circuit relates time complexity to circuit complexity.
Values
Whereas expressions denote mainly numbers in elementary algebra, in Boolean algebra, they denote the truth values false and true. These values are represented with the bits, 0 and 1. They do not behave like the integers 0 and 1, for which, but may be identified with the elements of the two-element field, that is, integer arithmetic modulo 2, for which. Addition and multiplication then play the Boolean roles of XOR and AND, respectively, with disjunction definable as and negation as. In, may be replaced by, since they denote the same operation; however, this way of writing Boolean operations allows applying the usual arithmetic operations of integers.Boolean algebra also deals with functions which have their values in the set. A sequence of bits is a commonly used example of such a function. Another common example is the totality of subsets of a set : to a subset of, one can define the indicator function that takes the value on, and outside. The most general example is the set elements of a Boolean algebra, with all of the foregoing being instances thereof.
As with elementary algebra, the purely equational part of the theory may be developed, without considering explicit values for the variables.
Operations
Basic operations
While elementary algebra has four operations, the Boolean algebra has only three basic operations: conjunction, disjunction, and negation, expressed with the corresponding binary operators AND and OR and the unary operator NOT, collectively referred to as Boolean operators. Variables in Boolean algebra that store the logical value of 0 and 1 are called the Boolean variables. They are used to store either true or false values. The basic operations on Boolean variables x and y are defined as follows:| Logical operation | Operator | Notation | Alternative notations | Definition |
| Conjunction | AND | |||
| Disjunction | OR | |||
| Negation | NOT | ¬x |
Alternatively, the values of,, and ¬x can be expressed by tabulating their values with truth tables as follows:
| 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 |
When used in expressions, the operators are applied according to the precedence rules. As with elementary algebra, expressions in parentheses are evaluated first, following the precedence rules.
If the truth values 0 and 1 are interpreted as integers, these operations may be expressed with the ordinary operations of arithmetic, or by the minimum/maximum functions:
One might consider that only negation and one of the two other operations are basic because of the following identities that allow one to define conjunction in terms of negation and the disjunction, and vice versa :
Secondary operations
Operations composed from the basic operations include, among others, the following:These definitions give rise to the following truth tables giving the values of these operations for all four possible inputs.
; Material conditional
; Exclusive OR
; '''Logical equivalence'''
Laws
A law of Boolean algebra is an identity such as between two Boolean terms, where a Boolean term is defined as an expression built up from variables and the constants 0 and 1 using the operations ∧, ∨, and ¬. The concept can be extended to terms involving other Boolean operations such as ⊕, →, and ≡, but such extensions are unnecessary for the purposes to which the laws are put. Such purposes include the definition of a Boolean algebra as any model of the Boolean laws, and as a means for deriving new laws from old as in the derivation of from .Monotone laws
Boolean algebra satisfies many of the same laws as ordinary algebra when one matches up ∨ with addition and ∧ with multiplication. In particular the following laws are common to both kinds of algebra:The following laws hold in Boolean algebra, but not in ordinary algebra:
Taking in the third law above shows that it is not an ordinary algebra law, since. The remaining five laws can be falsified in ordinary algebra by taking all variables to be 1. For example, in absorption law 1, the left hand side would be, while the right hand side would be 1.
All of the laws treated thus far have been for conjunction and disjunction. These operations have the property that changing either argument either leaves the output unchanged, or the output changes in the same way as the input. Equivalently, changing any variable from 0 to 1 never results in the output changing from 1 to 0. Operations with this property are said to be monotone. Thus the axioms thus far have all been for monotonic Boolean logic. Nonmonotonicity enters via complement ¬ as follows.
Nonmonotone laws
The complement operation is defined by the following two laws.All properties of negation including the laws below follow from the above two laws alone.
In both ordinary and Boolean algebra, negation works by exchanging pairs of elements, hence in both algebras it satisfies the double negation law
But whereas ordinary algebra satisfies the two laws
Boolean algebra satisfies De Morgan's laws: