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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
and
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, 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 study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, mathematics and
theoretical computer science Theoretical computer science is a subfield of computer science and mathematics that focuses on the Abstraction, abstract and mathematical foundations of computation. It is difficult to circumscribe the theoretical areas precisely. The Associati ...
, a Boolean domain is usually written as , or \mathbb. The
algebraic structure In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
that naturally builds on a Boolean domain is the Boolean algebra with two elements. The
initial object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element) ...
in the category of bounded lattices is a Boolean domain. In
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, a Boolean variable is a variable that takes values in some Boolean domain. Some
programming language A programming language is a system of notation for writing computer programs. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually def ...
s 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 data type in the strict sense. In C or
BASIC Basic or BASIC may refer to: Science and technology * BASIC, a computer programming language * Basic (chemistry), having the properties of a base * Basic access authentication, in HTTP Entertainment * Basic (film), ''Basic'' (film), a 2003 film ...
, 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 1-x, conjunction (AND) is replaced with multiplication (xy), and disjunction (OR) is defined via De Morgan's law to be 1-(1-x)(1-y)=x+y-xy. Interpreting these values as logical
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...
s yields a
multi-valued logic Many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values (i.e., "true" and "false") ...
, which forms the basis for
fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
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) (also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field with two elements. is the Field (mathematics), field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity ...


References


Further reading

*
https://m.tau.ac.il/~ilia1/publications/rpbd_book.pdf draft version -->
(xxx+428 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.) * (xxxv+1+445+1 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.) *

(xli+1+494 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 at the University of Bremen, Bremen, Germany on 2018-09-19/21.) * (vii+1+197+5 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.) * (xxii+231+1 pages) * {{cite book , editor-first1=Rolf , editor-last1=Drechsler , editor-link1=Rolf Drechsler , editor-first2=Sebastian , editor-last2=Huhn , title=Advanced Boolean Techniques - Selected Papers from the 15th International Workshop on Boolean Problems , publisher= Springer Nature Switzerland AG , publication-place=Cham, Switzerland , location=Bremen, Germany , edition=1 , date=2023-05-30 , isbn=978-3-031-28915-6 , doi=10.1007/978-3-031-28916-3 (viii+172+6 pages

(NB. Contains extended versions of the best manuscripts from the 15th International Workshop on Boolean Problems (IWSBP 2022) held at the University of Bremen, Bremen, Germany on 2022-09-22/23.) Boolean algebra