In
mathematics and
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, a Boolean domain is a
set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...
, mathematics and
theoretical computer science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory.
It is difficult to circumsc ...
, a Boolean domain is usually written as ,
or
The
algebraic structure that naturally builds on a Boolean domain is the
Boolean algebra with two elements. The
initial object in the
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
of
bounded lattices is a Boolean domain.
In
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includin ...
, a Boolean variable is a
variable that takes values in some Boolean domain. Some
programming languages feature
reserved words or symbols for the elements of the Boolean domain, for example
false
and
true
. However, many programming languages do not have a
Boolean datatype in the strict sense. In
C or
BASIC
BASIC (Beginners' All-purpose Symbolic Instruction Code) is a family of general-purpose, high-level programming languages designed for ease of use. The original version was created by John G. Kemeny and Thomas E. Kurtz at Dartmouth College ...
, for example, falsity is represented by the number 0 and truth is represented by the number 1 or −1, and all variables that can take these values can also take any other numerical values.
Generalizations
The Boolean domain can be replaced by the
unit interval , in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with
conjunction (AND) is replaced with multiplication (
), and disjunction (OR) is defined via
De Morgan's law to be
.
Interpreting these values as logical
truth values yields a
multi-valued logic, which forms the basis for
fuzzy logic and
probabilistic logic. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.
See also
*
Boolean-valued function
*
GF(2)
References
Further reading
*
(455 pages
(NB. Contains extended versions of the best manuscripts from the 10th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2012-09-19/21.)
* (480 pages
(NB. Contains extended versions of the best manuscripts from the 11th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2014-09-17/19.)
*
(536 pages
(NB. Contains extended versions of the best manuscripts from the 12th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2016-09-22/23.)
* (vii+265+7 pages
(NB. Contains extended versions of the best manuscripts from the 13th International Workshop on Boolean Problems (IWSBP 2018) held in Bremen, Germany on 2018-09-19/21.)
* {{cite book , editor-first1=Rolf , editor-last1=Drechsler , editor-link1=Rolf Drechsler , editor-first2=Daniel , editor-last2=Große , title=Recent Findings in Boolean Techniques - Selected Papers from the 14th International Workshop on Boolean Problems , publisher=
Springer Nature Switzerland AG
Springer Nature or the Springer Nature Group is a German-British academic publishing company created by the May 2015 merger of Springer Science+Business Media and Holtzbrinck Publishing Group's Nature Publishing Group, Palgrave Macmillan, and Macm ...
, edition=1 , date=2021-04-30 , isbn=978-3-030-68070-1 , doi= (204 pages
(NB. Contains extended versions of the best manuscripts from the 14th International Workshop on Boolean Problems (IWSBP 2020) held
COVID-19, virtually on 2020-09-24/25.)
Boolean algebra