In
logic,
mathematics and
linguistics, And (
) is the
truth-functional
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly one ...
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, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...
that represents this operator is typically written as
or .
is true if and only if
is true and
is true, otherwise it is false.
An operand of a conjunction is a conjunct.
Beyond logic, the term "conjunction" also refers to similar concepts in other fields:
* In
natural language, the
denotation of expressions such as
English "and".
* In
programming languages, the
short-circuit and control structure
In computer science, control flow (or flow of control) is the order in which individual statements, instructions or function calls of an imperative program are executed or evaluated. The emphasis on explicit control flow distinguishes an ''imp ...
.
* In
set theory,
intersection.
* In
lattice theory, logical conjunction (
greatest lower bound).
* 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 quantifi ...
,
universal quantification
In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other ...
.
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 His work centred on philosophical logic, mathematical logic and history of logic. ...
's
prefix notation for logic, the operator is K, for Polish ''koniunkcja''.
Definition
Logical conjunction is an
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
on two
logical values, typically the values of two
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s, that produces a value of ''true''
if and only if both of its operands are true.
The conjunctive
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), an ...
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 truth, when conjunction is defined as an operator or function of arbitrary
arity
Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathemati ...
, 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, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
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 is a classically
valid, simple
argument form
In logic, logical form of a Statement (logic), statement is a precisely-specified Semantics, semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly Syntactic ambiguity, ambiguous sta ...
. 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 notation:
:
:
:
Here is an example of an argument that fits the form ''
conjunction introduction'':
:Bob likes apples.
:Bob likes oranges.
:Therefore, Bob likes apples and Bob likes oranges.
Conjunction elimination
In propositional logic, conjunction elimination (also called ''and'' elimination, ∧ elimination, or simplification)Hurley is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction ' ...
is another classically
valid, simple
argument form
In logic, logical form of a Statement (logic), statement is a precisely-specified Semantics, semantic version of that statement in a formal system. Informally, the logical form attempts to formalize a possibly Syntactic ambiguity, ambiguous sta ...
. 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 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, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name ...
: 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 valid rule of replacement ...
: yes
distributivity: with various operations, especially with ''
or''
idempotency
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of p ...
: yes
monotonicity: 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: 1 (the function is
bent)
If using
binary
Binary may refer to:
Science and technology Mathematics
* Binary number, a representation of numbers using only two digits (0 and 1)
* Binary function, a function that takes two arguments
* Binary operation, a mathematical operation that ta ...
values for true (1) and false (0), then ''logical conjunction'' works exactly like normal arithmetic
multiplication.
Applications in computer engineering
In high-level computer programming and
digital electronics, 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 that allows a current to travel along an unintended path with no or very low electrical impedance. This results in an excessive current flowing through the circuit. ...
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 viewed as
bitstring
A bit array (also known as bitmask, bit map, bit set, bit string, or bit vector) is an array data structure that compactly stores bits. It can be used to implement a simple set data structure. A bit array is effective at exploiting bit-level pa ...
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
In computer science, a mask or bitmask is data that is used for bitwise operations, particularly in a bit field. Using a mask, multiple bits in a byte, nibble, word, etc. can be set either on or off, or inverted from on to off (or vice versa) ...
. 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 computers sharing resources located on or provided by network nodes. The computers use common communication protocols over digital interconnections to communicate with each other. These interconnections are m ...
, bit masks are used to derive the network address of a
subnet 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 that uses the Internet Protocol for communication.. Updated by . An IP address serves two main functions: network interface ident ...
, by ANDing the IP address and the
subnet mask.
Logical conjunction "
AND
" is also used in
SQL operations to form
database queries.
The
Curry–Howard correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relati ...
relates logical conjunction to
product types.
Set-theoretic correspondence
The membership of an element of an
intersection set in
set theory 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 valid rule of replacement ...
,
commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name ...
and
idempotence.
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 ''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 that implements logical conjunction (∧) from mathematical logic AND gate behaves according to the truth table. A HIGH output (1) results only if all the inputs to the AND gate are HIGH (1). If not all ...
*
Bitwise AND
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic opera ...
*
Boolean algebra (logic)
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
*
Boolean algebra topics
*
Boolean conjunctive query In the theory of relational databases, a Boolean conjunctive query is a conjunctive query without distinguished predicates, i.e., a query in the form R_1(t_1) \wedge \cdots \wedge R_n(t_n), where each R_i is a relation symbol and each t_i is a tupl ...
*
Boolean domain
In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written ...
*
Boolean function
In mathematics, a Boolean function is a function whose arguments and result assume values from a two-element set (usually , or ). Alternative names are switching function, used especially in older computer science literature, and truth functio ...
*
Boolean-valued function
A Boolean-valued function (sometimes called a predicate or a proposition) is a function of the type f : X → B, where X is an arbitrary set and where B is a Boolean domain, i.e. a generic two-element set, (for example B = ), whose elements are ...
*
Conjunction elimination
In propositional logic, conjunction elimination (also called ''and'' elimination, ∧ elimination, or simplification)Hurley is a valid immediate inference, argument form and rule of inference which makes the inference that, if the conjunction ' ...
*
De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...
*
First-order 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 quantifi ...
*
Fréchet inequalities In probabilistic logic, the Fréchet inequalities, also known as the Boole–Fréchet inequalities, are rules implicit in the work of George BooleBoole, G. (1854). ''An Investigation of the Laws of Thought, On Which Are Founded the Mathematical Theo ...
*
Grammatical conjunction
*
Logical disjunction
In logic, disjunction is a logical connective typically notated as \lor and read aloud as "or". For instance, the English language sentence "it is raining or it is snowing" can be represented in logic using the disjunctive formula R \lor S ...
*
Logical negation
*
Logical graph
*
Operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
*
Peano–Russell notation In mathematical logic, Peano–Russell notation was Bertrand Russell's application of Giuseppe Peano's logical notation to the logical notions of Frege and was used in the writing of ''Principia Mathematica'' in collaboration with Alfred North White ...
*
Propositional calculus
References
External links
*
Wolfram MathWorld: Conjunction*
{{Authority control
Conjunction
Conjunction may refer to:
* Conjunction (grammar), a part of speech
* Logical conjunction, a mathematical operator
** Conjunction introduction, a rule of inference of propositional logic
* Conjunction (astronomy), in which two astronomical bodies ...
Semantics