
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 (
) 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
or .
is true if and only if
is true and
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
,
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
:
Defined by other operators
In systems where logical conjunction is not a primitive, it may be defined as
:
or
:
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:
:
:
:
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:
:
:
...or alternatively,
:
:
Negation
Definition
A conjunction
is proven false by establishing either
or
. In terms of the object language, this reads
:
This formula can be seen as a special case of
:
when
is a false proposition.
Other proof strategies
If
implies
, then both
as well as
prove the conjunction false:
:
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
:
when
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.
See also
*
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 ...
References
External links
*
Wolfram MathWorld: Conjunction*
{{Authority control
Conjunction
Conjunction may refer to:
* Conjunction (astronomy), in which two astronomical bodies appear close together in the sky
* Conjunction (astrology), astrological aspect in horoscopic astrology
* Conjunction (grammar), a part of speech
* Logical conjun ...
Semantics