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# Boolean algebra

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. 

## A New Kind of Science

A New Kind of Science is a best-selling, controversial book by Stephen Wolfram, published by his own company in 2002.

## A Symbolic Analysis of Relay and Switching Circuits

A Symbolic Analysis of Relay and Switching Circuits is the title of a master's thesis written by computer science pioneer Claude E. Shannon while attending the Massachusetts Institute of Technology (MIT) in 1937.

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

## Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.

## Algebra of sets

The algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion.

## Algebraic logic

In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.

## Algebraic semantics (mathematical logic)

In mathematical logic, algebraic semantics is a formal semantics based on algebras studied as part of algebraic logic.

## Algebraic structure

In mathematics, and more specifically in abstract algebra, an algebraic structure on a set A (called carrier set or underlying set) is a collection of finitary operations on A; the set A with this structure is also called an algebra.

## Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

## Article (grammar)

An article (with the linguistic glossing abbreviation) is a word that is used with a noun (as a standalone word or a prefix or suffix) to specify grammatical definiteness of the noun, and in some languages extending to volume or numerical scope.

## Assembly language

An assembly (or assembler) language, often abbreviated asm, is a low-level programming language, in which there is a very strong (but often not one-to-one) correspondence between the assembly program statements and the architecture's machine code instructions.

## Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

## Axiom

An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.

## Axiom of choice

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.

## Axiomatic system

In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems.

## Binary decision diagram

In computer science, a binary decision diagram (BDD) or branching program is a data structure that is used to represent a Boolean function.

## Binary number

In mathematics and digital electronics, a binary number is a number expressed in the base-2 numeral system or binary numeral system, which uses only two symbols: typically 0 (zero) and 1 (one).

## Bit

The bit (a portmanteau of binary digit) is a basic unit of information used in computing and digital communications.

## Bit array

A bit array (also known as bit map, bit set, bit string, or bit vector) is an array data structure that compactly stores bits.

## Bit blit

Bit blit (also written BITBLT, BIT BLT, BitBLT, Bit BLT, Bit Blt etc., which stands for bit block transfer) is a data operation commonly used in computer graphics in which several bitmaps are combined into one using a boolean function.

## Bitwise operation

In digital computer programming, a bitwise operation operates on one or more bit patterns or binary numerals at the level of their individual bits.

## Boolean algebra

In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively.

## Boolean algebra (structure)

In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice.

## Boolean algebras canonically defined

Boolean algebra is a mathematically rich branch of abstract algebra.

## Boolean circuit

In computational complexity theory and circuit complexity, a Boolean circuit is a mathematical model for digital logic circuits.

## Boolean differential calculus

Boolean differential calculus (BDC) (German: Boolescher Differentialkalkül (BDK)) is a subject field of Boolean algebra discussing changes of Boolean variables and Boolean functions.

## Boolean function

In mathematics and logic, a (finitary) Boolean function (or switching function) is a function of the form ƒ: Bk → B, where B.

## Boolean prime ideal theorem

In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given algebra.

## Boolean satisfiability problem

In computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated as SATISFIABILITY or SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula.

## Booleo

Booleo (stylized bOOleO) is a strategy card game using boolean logic gates.

## Carry (arithmetic)

In elementary arithmetic, a carry is a digit that is transferred from one column of digits to another column of more significant digits.

## Charles Sanders Peirce

Charles Sanders Peirce ("purse"; 10 September 1839 – 19 April 1914) was an American philosopher, logician, mathematician, and scientist who is sometimes known as "the father of pragmatism".

## Circuit complexity

In theoretical computer science, circuit complexity is a branch of computational complexity theory in which Boolean functions are classified according to the size or depth of Boolean circuits that compute them.

## Circuit diagram

A circuit diagram (electrical diagram, elementary diagram, electronic schematic) is a graphical representation of an electrical circuit.

## Claude Shannon

Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, and cryptographer known as "the father of information theory".

## Cofiniteness

In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set.

## Combinational logic

In digital circuit theory, combinational logic (sometimes also referred to as time-independent logic) is a type of digital logic which is implemented by Boolean circuits, where the output is a pure function of the present input only.

## Complemented lattice

In the mathematical discipline of order theory, a complemented lattice is a bounded lattice (with least element 0 and greatest element 1), in which every element a has a complement, i.e. an element b satisfying a ∨ b.

## Computer

A computer is a device that can be instructed to carry out sequences of arithmetic or logical operations automatically via computer programming.

## Computer monitor

A computer monitor is an output device which displays information in pictorial form.

## Computer-aided design

Computer-aided design (CAD) is the use of computer systems to aid in the creation, modification, analysis, or optimization of a design.

## De Morgan's laws

In propositional logic and boolean algebra, De Morgan's laws are a pair of transformation rules that are both valid rules of inference.

## Decision problem

In computability theory and computational complexity theory, a decision problem is a problem that can be posed as a yes-no question of the input values.

## Digital electronics

Digital electronics or digital (electronic) circuits are electronics that operate on digital signals.

## Distributive lattice

In mathematics, a distributive lattice is a lattice in which the operations of join and meet distribute over each other.

## Edward Vermilye Huntington

Edward Vermilye Huntington (April 26, 1874, Clinton, New York, USANovember 25, 1952, Cambridge, Massachusetts, USA) was an American mathematician.

## Electronic design automation

Electronic design automation (EDA), also referred to as electronic computer-aided design (ECAD), is a category of software tools for designing electronic systems such as integrated circuits and printed circuit boards.

## Elementary algebra

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics.

## Ernst Schröder

Friedrich Wilhelm Karl Ernst Schröder (25 November 1841 in Mannheim, Baden, Germany &ndash; 16 June 1902 in Karlsruhe, Germany) was a German mathematician mainly known for his work on algebraic logic.

## Exclusive or

Exclusive or or exclusive disjunction is a logical operation that outputs true only when inputs differ (one is true, the other is false).

## False premise

A false premise is an incorrect proposition that forms the basis of an argument or syllogism.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Field of sets

In mathematics a field of sets is a pair \langle X, \mathcal \rangle where X is a set and \mathcal is an algebra over X i.e., a non-empty subset of the power set of X closed under the intersection and union of pairs of sets and under complements of individual sets.

## First-order logic

First-order logic—also known as first-order predicate calculus and predicate logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.

## Formal system

A formal system is the name of a logic system usually defined in the mathematical way.

## Formal verification

In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of intended algorithms underlying a system with respect to a certain formal specification or property, using formal methods of mathematics.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.

## Fuzzy logic

Fuzzy logic is a form of many-valued logic in which the truth values of variables may be any real number between 0 and 1.

## George Boole

George Boole (2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland.

## GF(2)

GF(2) (also F2, Z/2Z or Z2) is the '''G'''alois '''f'''ield of two elements.

Google LLC is an American multinational technology company that specializes in Internet-related services and products, which include online advertising technologies, search engine, cloud computing, software, and hardware.

Google Scholar is a freely accessible web search engine that indexes the full text or metadata of scholarly literature across an array of publishing formats and disciplines.

## Greatest common divisor

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

## Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

## Henry M. Sheffer

Henry Maurice Sheffer (September 1, 18821964) was an American logician.

## Heyting algebra

In mathematics, a Heyting algebra is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation a → b of implication such that c ∧ a ≤ b is equivalent to c ≤ a → b. From a logical standpoint, A → B is by this definition the weakest proposition for which modus ponens, the inference rule A → B, A ⊢ B, is sound.

## Hilbert system

In logic, especially mathematical logic, a Hilbert system, sometimes called Hilbert calculus, Hilbert-style deductive system or Hilbert&ndash;Ackermann system, is a type of system of formal deduction attributed to Gottlob FregeMáté & Ruzsa 1997:129 and David Hilbert.

## Idempotence

Idempotence is the property of certain operations in mathematics and computer science that they can be applied multiple times without changing the result beyond the initial application.

## Identity (mathematics)

In mathematics an identity is an equality relation A.

## Index set

In mathematics, an index set is a set whose members label (or index) members of another set.

## Indexed family

In mathematics, an indexed family is informally a collection of objects, each associated with an index from some index set.

## Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Interpretation (logic)

An interpretation is an assignment of meaning to the symbols of a formal language.

## Intersection (set theory)

In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

## Intuitionistic logic

Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Józef Maria Bocheński

Józef Maria Bocheński (Czuszów, Congress Poland, Russian Empire, 30 August 1902 – 8 February 1995, Fribourg, Switzerland) was a Polish Dominican, logician and philosopher.

## John Etchemendy

John W. Etchemendy (born 1952 in Reno, Nevada) was Stanford University's twelfth Provost.

## Jon Barwise

Kenneth Jon Barwise (June 29, 1942 – March 5, 2000) was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used.

## Klein four-group

In mathematics, the Klein four-group (or just Klein group or Vierergruppe, four-group, often symbolized by the letter V or as K4) is the group, the direct product of two copies of the cyclic group of order 2.

## Least common multiple

In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero.

## List of Boolean algebra topics

This is a list of topics around Boolean algebra and propositional logic.

## Logic gate

In electronics, a logic gate is an idealized or physical device implementing a Boolean function; that is, it performs a logical operation on one or more binary inputs and produces a single binary output.

## Logic optimization

Logic optimization, a part of logic synthesis in electronics, is the process of finding an equivalent representation of the specified logic circuit under one or more specified constraints.

## Logic synthesis

In electronics, logic synthesis is a process by which an abstract form of desired circuit behavior, typically at register transfer level (RTL), is turned into a design implementation in terms of logic gates, typically by a computer program called a synthesis tool.

## Logical conjunction

In logic, mathematics and linguistics, And (∧) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true.

## Logical connective

In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a symbol or word used to connect two or more sentences (of either a formal or a natural language) in a grammatically valid way, such that the value of the compound sentence produced depends only on that of the original sentences and on the meaning of the connective.

## Logical consequence

Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements.

## Logical disjunction

In logic and mathematics, or is the truth-functional operator of (inclusive) disjunction, also known as alternation; the or of a set of operands is true if and only if one or more of its operands is true.

## Machine code

Machine code is a computer program written in machine language instructions that can be executed directly by a computer's central processing unit (CPU).

## Many-valued logic

In logic, a many-valued logic (also multi- or multiple-valued logic) is a propositional calculus in which there are more than two truth values.

## Marshall Harvey Stone

Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras.

In computer science, a mask is data that is used for bitwise operations, particularly in a bit field.

## Mathematical logic

Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.

## Mathematical structure

In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Model of computation

In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how a set of outputs are computed given a set of inputs.

## Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

## Naive set theory

Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics.

## Negation

In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P (¬P), which is interpreted intuitively as being true when P is false, and false when P is true.

## NP-completeness

In computational complexity theory, an NP-complete decision problem is one belonging to both the NP and the NP-hard complexity classes.

## Number

A number is a mathematical object used to count, measure and also label.

## Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

## Partition of a set

In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets.

## Philosophical Magazine

The Philosophical Magazine is one of the oldest scientific journals published in English.

## Pixel

In digital imaging, a pixel, pel, dots, or picture element is a physical point in a raster image, or the smallest addressable element in an all points addressable display device; so it is the smallest controllable element of a picture represented on the screen.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Power set

In mathematics, the power set (or powerset) of any set is the set of all subsets of, including the empty set and itself, variously denoted as, 𝒫(), ℘() (using the "Weierstrass p"),,, or, identifying the powerset of with the set of all functions from to a given set of two elements,.

## Principle of bivalence

In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false.

## Probabilistic logic

The aim of a probabilistic logic (also probability logic and probabilistic reasoning) is to combine the capacity of probability theory to handle uncertainty with the capacity of deductive logic to exploit structure of formal argument.

## Programming language

A programming language is a formal language that specifies a set of instructions that can be used to produce various kinds of output.

## Propositional calculus

Propositional calculus is a branch of logic.

## Propositional formula

In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value.

## Propositional variable

In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is a variable which can either be true or false.

## Quantifier (logic)

In logic, quantification specifies the quantity of specimens in the domain of discourse that satisfy an open formula.

## Raster graphics

In computer graphics, a raster graphics or bitmap image is a dot matrix data structure that represents a generally rectangular grid of pixels (points of color), viewable via a monitor, paper, or other display medium.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Relation algebra

In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation.

## Relevance logic

Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications to be relevantly related.

## Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

## Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

## Sequent

In mathematical logic, a sequent is a very general kind of conditional assertion.

## Sequent calculus

Sequent calculus is, in essence, a style of formal logical argumentation where every line of a proof is a conditional tautology (called a sequent by Gerhard Gentzen) instead of an unconditional tautology.

## Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects.

## Sheffer stroke

In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written ↑, also written | (not to be confused with "||", which is often used to represent disjunction), or Dpq (in Bocheński notation), denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both".

## Solid modeling

Solid modeling (or modelling) is a consistent set of principles for mathematical and computer modeling of three-dimensional solids.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Square-free integer

In mathematics, a square-free integer is an integer which is divisible by no perfect square other than 1.

## Stanford Encyclopedia of Philosophy

The Stanford Encyclopedia of Philosophy (SEP) combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users.

## Statistics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.

## Stephen Wolfram

Stephen Wolfram (born August 29, 1959) is a British-American computer scientist, physicist, and businessman.

## Stone's representation theorem for Boolean algebras

In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets.

## Structure (mathematical logic)

In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.

## Switching circuit theory

Switching circuit theory is the mathematical study of the properties of networks of idealized switches.

## Tautology (logic)

In logic, a tautology (from the Greek word ταυτολογία) is a formula or assertion that is true in every possible interpretation.

## Taylor & Francis

Taylor & Francis Group is an international company originating in England that publishes books and academic journals.

## The Laws of Thought

An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities by George Boole, published in 1854, is the second of Boole's two monographs on algebraic logic.

## Theoretical computer science

Theoretical computer science, or TCS, is a subset of general computer science and mathematics that focuses on more mathematical topics of computing and includes the theory of computation.

## Three-valued logic

In logic, a three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which there are three truth values indicating true, false and some indeterminate third value.

## Time complexity

In computer science, the time complexity is the computational complexity that describes the amount of time it takes to run an algorithm.

## Truth table

A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional arguments, that is, for each combination of values taken by their logical variables (Enderton, 2001).

## 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.

## Two-element Boolean algebra

In mathematics and abstract algebra, the two-element Boolean algebra is the Boolean algebra whose underlying set (or universe or carrier) B is the Boolean domain.

## Uncountable set

In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.

## Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

## Unit interval

In mathematics, the unit interval is the closed interval, that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1.

## Variable (mathematics)

In elementary mathematics, a variable is a symbol, commonly an alphabetic character, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown.

## Vector logic

Vector logicMizraji, E. (1992).

## Venn diagram

A Venn diagram (also called primary diagram, set diagram or logic diagram) is a diagram that shows all possible logical relations between a finite collection of different sets.

## Very-large-scale integration

Very-large-scale integration (VLSI) is the process of creating an integrated circuit (IC) by combining hundreds of thousands of transistors or devices into a single chip.

## Video card

A video card (also called a display card, graphics card, display adapter or graphics adapter) is an expansion card which generates a feed of output images to a display (such as a computer monitor).

## Voxel

A voxel represents a value on a regular grid in three-dimensional space.

## Walter Gottschalk

Walter Helbig Gottschalk (November 3, 1918 – February 15, 2004) was an American mathematician, one of the founders of topological dynamics.

## William Stanley Jevons

William Stanley Jevons FRS (1 September 1835 – 13 August 1882) was an English economist and logician.

## Word (computer architecture)

In computing, a word is the natural unit of data used by a particular processor design.

## References

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