TheInfoList

In
logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents statements and ar ...

,
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
and
linguistics Linguistics is the scientific study of language A language is a structured system of communication Communication (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo ...

, And ($\wedge$) 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. The
logical connective In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
that represents this operator is typically written as $\wedge$ or . $A \land B$ is true if and only if $A$ is true and $B$ is true. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In
natural language In neuropsychology Neuropsychology is a branch of psychology. It is concerned with how a person's cognition and behavior are related to the brain and the rest of the nervous system. Professionals in this branch of psychology often focus on ...
, the
denotation The denotation of a word is its central sense A sense is a biological system A biological system is a complex biological network, network which connects several biologically relevant entities. Biological organization spans several scales and ar ...
of expressions such as
English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval England, which has eventually become the World language, leading lan ...

"and". * In
programming language A programming language is a formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

s, the short-circuit and
control structure In computer science, control flow (or flow of control) is the order in which individual Statement (computer science), statements, Instruction (computer science), instructions or function calls of an imperative programming, imperative computer prog ...
. * In
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
,
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
. * In
lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every two elements have a unique supremum (also called a least upper boun ...
, logical conjunction (
greatest lower bound are equal. Image:Supremum illustration.svg, 250px, A set ''A'' of real numbers (blue circles), a set of upper bounds of ''A'' (red diamond and circles), and the smallest such upper bound, that is, the supremum of ''A'' (red diamond). In mathematic ...
). * In
predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses Quantifica ...
,
universal quantification In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical ...
.

# Notation

And is usually denoted by an infix operator: in mathematics and logic, it is denoted by $\wedge$, or ; in electronics, ; and in programming languages, &, &&, or and. In
Jan Łukasiewicz Jan Łukasiewicz (; 21 December 1878 – 13 February 1956) was a Polish logician and philosopher who is best known for Polish notation and Łukasiewicz logic. He was born in Lemberg, a city in the Austrian Galicia, Galician Kingdom of Austria-Hungar ...

's prefix notation for logic, the operator is K, for Polish ''koniunkcja''.

# Definition

Logical conjunction is an operation on two
logical value In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...
s, typically the values of two
proposition In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, lab ...
s, that produces a value of ''true''
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
both of its operands are true. The conjunctive
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
is true, which is to say that AND-ing an expression with true will never change the value of the expression. In keeping with the concept of
vacuous truthIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, when conjunction is defined as an operator or function of arbitrary
arity Arity () is the number of arguments In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logo ...
, the empty conjunction (AND-ing over an empty set of operands) is often defined as having the result true.

## Truth table

The
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra (logic), Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expression (mathematics) ...

of $A \land B$:

## Defined by other operators

In systems where logical conjunction is not a primitive, it may be defined as :$A \land B = \neg\left(A \to \neg B\right)$ or :$A \land B = \neg\left(\neg A \lor \neg B\right).$

# Introduction and elimination rules

As a rule of inference,
conjunction introduction Conjunction introduction (often abbreviated simply as conjunction and also called and introduction)Moore and Parker is a Validity (logic), valid rule of inference of propositional calculus, propositional logic. The rule makes it possible to introdu ...
is a classically valid, simple
argument form In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument ...
. The argument form has two premises, ''A'' and ''B''. Intuitively, it permits the inference of their conjunction. :''A'', :''B''. :Therefore, ''A'' and ''B''. or in
logical operator Logic (from Greek: grc, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argument In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason ...
notation: :$A,$ :$B$ :$\vdash A \land B$ Here is an example of an argument that fits the form ''
conjunction introduction Conjunction introduction (often abbreviated simply as conjunction and also called and introduction)Moore and Parker is a Validity (logic), valid rule of inference of propositional calculus, propositional logic. The rule makes it possible to introdu ...
'': :Bob likes apples. :Bob likes oranges. :Therefore, Bob likes apples and Bob likes oranges.
Conjunction elimination In propositional calculus, propositional logic, conjunction elimination (also called ''and'' elimination, ∧ elimination, or simplification)Hurley is a Validity (logic), valid immediate inference, argument form and rule of inference which makes th ...
is another classically valid, simple
argument form In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument ...
. Intuitively, it permits the inference from any conjunction of either element of that conjunction. :''A'' and ''B''. :Therefore, ''A''. ...or alternatively, :''A'' and ''B''. :Therefore, ''B''. In
logical operator Logic (from Greek: grc, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argument In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason ...
notation: :$A \land B$ :$\vdash A$ ...or alternatively, :$A \land B$ :$\vdash B$

# Negation

## Definition

A conjunction $A\land B$ is proven false by establishing either $\neg A$ or $\neg B$. In terms of the object language, this reads :$\neg A\to\neg\left(A\land B\right)$ This formula can be seen as a special case of :$\left(A\to C\right) \to \left( \left(A\land B\right)\to C \right)$ when $C$ is a false proposition.

## Other proof strategies

If $A$ implies $\neg B$, then both $\neg A$ as well as $A$ prove the conjunction false: :$\left(A\to\negB\right)\to\neg\left(A\land B\right)$ In other words, a conjunction can actually be proven false just by knowing about the relation of its conjuncts, and not necessary about their truth values. This formula can be seen as a special case of :$\left(A\to\left(B\to C\right)\right)\to \left( \left(A\land B\right)\to C \right)$ when $C$ is a false proposition. Either of the above are constructively valid proofs by contradiction.

# Properties

commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
: yes
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
: yes
distributivity In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: with various operations, especially with '' or''
idempotency Idempotence (, ) is the property of certain operations in mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
: yes
monotonicity Figure 3. A function that is not monotonic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...
: yes truth-preserving: yes
When all inputs are true, the output is true. falsehood-preserving: yes
When all inputs are false, the output is false. Walsh spectrum: (1,-1,-1,1) Non
linearity Linearity is the property of a mathematical relationship (''function (mathematics), function'') that can be graph of a function, graphically represented as a straight Line (geometry), line. Linearity is closely related to ''Proportionality (math ...
: 1 (the function is bent) If using
binary Binary may refer to: Science and technology Mathematics * 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: ty ...
values for true (1) and false (0), then ''logical conjunction'' works exactly like normal arithmetic
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

.

# Applications in computer engineering

In high-level computer programming and
digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons The electron is a subatomic particle In physica ...
, logical conjunction is commonly represented by an infix operator, usually as a keyword such as "AND", an algebraic multiplication, or the ampersand symbol & (sometimes doubled as in &&). Many languages also provide
short-circuit A short circuit (sometimes abbreviated to short or s/c) is an electrical circuit An electrical network is an interconnection of electrical component An electronic component is any basic discrete device or physical entity in an electron ...
control structures corresponding to logical conjunction. Logical conjunction is often used for bitwise operations, where 0 corresponds to false and 1 to true: * 0 AND 0  =  0, * 0 AND 1  =  0, * 1 AND 0  =  0, * 1 AND 1  =  1. The operation can also be applied to two binary
words In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most lang ...

viewed as
bitstring A bit array (also known as bit map, bit set, bit string, or bit vector) is an array data structure ARRAY, also known as ARRAY Now, is an independent distribution company launched by film maker and former publicist Ava DuVernay Ava Marie DuVe ...
s of equal length, by taking the bitwise AND of each pair of bits at corresponding positions. For example: * 11000110 AND 10100011  =  10000010. This can be used to select part of a bitstring using a bit mask. For example, 10011101 AND 00001000  =  00001000 extracts the fifth bit of an 8-bit bitstring. In
computer networking A computer network is a set of computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern computers can perform generic sets of operations known as Computer ...
, bit masks are used to derive the network address of a
subnet A subnetwork or subnet is a logical subdivision of an IP network. Updated by RFC 6918. The practice of dividing a network into two or more networks is called subnetting. Computers that belong to the same subnet are addressed with an identical ...
within an existing network from a given
IP address An Internet Protocol address (IP address) is a numerical label such as that is connected to a computer network A computer network is a set of s sharing resources located on or provided by . The computers use common s over to communi ...
, by ANDing the IP address and the
subnet mask A subnetwork or subnet is a logical subdivision of an IP network The Internet protocol suite, commonly known as TCP/IP, is the set of communications protocol A communication protocol is a system of rules that allows two or more entit ...
. Logical conjunction "AND" is also used in
SQL SQL ( ''S-Q-L'', "sequel"; Structured Query Language) is a domain-specific languageA domain-specific language (DSL) is a computer languageA computer language is a method of communication with a computer A computer is a machine that can b ...

operations to form
database In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and sof ...

queries. The
Curry–Howard correspondence In programming language theory and proof theory Proof may refer to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Formal sciences * Formal proof, a c ...
relates logical conjunction to
product type In programming language A programming language is a formal language comprising a Instruction set architecture, set of instructions that produce various kinds of Input/output, output. Programming languages are used in computer programming to impl ...
s.

# Set-theoretic correspondence

The membership of an element of an intersection set in
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, i ...
is defined in terms of a logical conjunction: ''x'' ∈ ''A'' ∩ ''B'' if and only if (''x'' ∈ ''A'') ∧ (''x'' ∈ ''B''). Through this correspondence, set-theoretic intersection shares several properties with logical conjunction, such as
associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule ...
,
commutativity In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

and
idempotence Idempotence (, ) is the property of certain operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, m ...
.

# Natural language

As with other notions formalized in mathematical logic, the logical conjunction ''and'' is related to, but not the same as, the
grammatical conjunction In grammar In linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as ...
''and'' in natural languages. English "and" has properties not captured by logical conjunction. For example, "and" sometimes implies order having the sense of "then". For example, "They got married and had a child" in common discourse means that the marriage came before the child. The word "and" can also imply a partition of a thing into parts, as "The American flag is red, white, and blue." Here, it is not meant that the flag is ''at once'' red, white, and blue, but rather that it has a part of each color.

* And-inverter graph *
AND gate The AND gate is a basic digital logic gate A logic gate is an idealized model of computation A model is an informative representation of an object, person or system. The term originally denoted the plan A plan is typically any diagram or l ...

*
Bitwise AND In computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, gen ...
*
Boolean algebra (logic) In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, respectively. Instead of elementary a ...
* Boolean algebra topics *
Boolean conjunctive queryIn the theory of relational databases A relational database is a digital database based on the relational model of data, as proposed by E. F. Codd in 1970. A software system used to maintain relational databases is a relational database manageme ...
*
Boolean domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
*
Boolean function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
*
Boolean-valued function A Boolean-valued function (sometimes called a predicate or a proposition In linguistics and logic, a proposition is the meaning of a declarative sentence. In philosophy, "Meaning (philosophy), meaning" is understood to be a non-linguistic enti ...
*
Conjunction elimination In propositional calculus, propositional logic, conjunction elimination (also called ''and'' elimination, ∧ elimination, or simplification)Hurley is a Validity (logic), valid immediate inference, argument form and rule of inference which makes th ...
*
De Morgan's laws In propositional calculus, propositional logic and Boolean algebra, De Morgan's laws are a pair of transformation rules that are both Validity (logic), valid rule of inference, rules of inference. They are named after Augustus De Morgan, a 19th ...
*
First-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rul ...
*
Fréchet inequalitiesIn probabilistic logicThe 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 pr ...
*
Grammatical conjunction In grammar In linguistics Linguistics is the scientific study of language, meaning that it is a comprehensive, systematic, objective, and precise study of language. Linguistics encompasses the analysis of every aspect of language, as ...
*
Logical disjunction In logic, disjunction is a logical connective typically notated \lor whose meaning either refines or corresponds to that of natural language expressions such as "or". In classical logic, it is given a truth functional semantics of logic, seman ...
*
Logical negation In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argume ...
* Logical graph * Operation * Peano–Russell notation *
Propositional calculus Propositional calculus is a branch of logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fal ...