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| {{redirect|∧|the logic gate|AND gate|exterior product|Exterior algebra}}
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| [[File:Venn0001.svg|220px|thumb|[[Venn diagram]] of <math>\scriptstyle A \and B</math>]]
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| [[File:Venn 0000 0001.svg|220px|thumb|Venn diagram of <math>\scriptstyle A \and B \and C</math>]]
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| In [[logic]] and [[mathematics]], a two-place [[logical operator]] '''and''', also known as '''logical conjunction''',<ref>Moore and Parker, ''Critical Thinking''</ref> results in ''true'' if both of its [[operands]] are true, otherwise the value of ''false''.
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| The analogue of conjunction for a (possibly [[Infinity|infinite]]) family of statements is [[universal quantification]], which is part of [[predicate logic]].
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| ==Notation==
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| '''And''' is usually expressed with the prefix operator '''K''', or an infix operator. In mathematics and logic, the infix operator is usually '''∧'''; in electronics '''<math>\cdot</math>'''; and in programming languages, '''&''' or '''and'''. Some programming languages have a related [[control structure]], the [[short-circuit evaluation|short-circuit and]], written '''&&''', '''and then''', etc.
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| ==Definition==
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| '''Logical conjunction''' is an [[logical operation|operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''true'' [[Iff|if and only if]] both of its operands are true.
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| The conjunctive [[identity element|identity]] is 1, which is to say that AND-ing an expression with 1 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]], the empty conjunction (AND-ing over an empty set of operands) is often defined as having the result 1.
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| ===Truth table===
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| [[File:Multigrade operator AND.svg|thumb|Conjunctions of the arguments on the left
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| >The [[truth value|true]] [[bit]]<nowiki>s</nowiki> form a [[Sierpinski triangle]]]]
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| The [[truth table]] of <math>~A \and B</math>:
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| {| class="wikitable" style="margin: 0 0 1em 1em"
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| |- bgcolor="#ddeeff" align="center"
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| |colspan=2|'''INPUT''' || '''OUTPUT'''
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| |- bgcolor="#ddeeff" align="center"
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| | <math> A</math>|| <math>B</math> || <math> A \and B</math>
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| |- bgcolor="#ddffdd" align="center"
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| |T || T || T
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| |- bgcolor="#ddffdd" align="center"
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| |T || F || F
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| |- bgcolor="#ddffdd" align="center"
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| |F || T || F
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| |- bgcolor="#ddffdd" align="center"
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| |F || F || F
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| |}
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| ==Introduction and elimination rules==
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| As a rule of inference, conjunction introduction is a classically [[validity|valid]], simple [[argument form]]. The argument form has two premises, ''A'' and ''B''. Intuitively, it permits the inference of their conjunction. | |
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| :''A'',
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| :''B''.
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| :Therefore, ''A'' and ''B''.
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| or in [[logical operator]] notation:
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| :<math> A, </math>
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| :<math> B </math>
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| :<math> \vdash A \and B </math>
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| Here is an example of an argument that fits the form ''[[conjunction introduction]]'':
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| :Bob likes apples.
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| :Bob likes oranges.
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| :Therefore, Bob likes apples and oranges.
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| [[Conjunction elimination]] is another classically [[validity|valid]], simple [[argument form]]. Intuitively, it permits the inference from any conjunction of either element of that conjunction.
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| :''A'' and ''B''.
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| :Therefore, ''A''.
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| ...or alternately,
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| :''A'' and ''B''.
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| :Therefore, ''B''.
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| In [[logical operator]] notation:
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| :<math> A \and B </math>
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| :<math> \vdash A </math>
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| ...or alternately, | |
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| :<math> A \and B </math>
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| :<math> \vdash B </math>
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| ==Properties==
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| '''[[commutativity]]: yes'''
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| {| style="text-align: center; border: 1px solid darkgray;"
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| |-
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| |<math>A \and B</math>
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| | <math>\Leftrightarrow</math>
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| |<math>B \and A</math>
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| |-
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| |[[File:Venn0001.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn0001.svg|50px]]
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| |}
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| '''[[associativity]]: yes'''
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| {| style="text-align: center; border: 1px solid darkgray;"
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| |-
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| |<math>~A</math>
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| |<math>~~~\and~~~</math>
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| |<math>(B \and C)</math>
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| | <math>\Leftrightarrow</math>
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| |<math>(A \and B)</math>
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| |<math>~~~\and~~~</math>
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| |<math>~C</math>
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| |-
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| |[[File:Venn 0101 0101.svg|50px]]
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| |<math>~~~\and~~~</math>
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| |[[File:Venn 0000 0011.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn 0000 0001.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn 0001 0001.svg|50px]]
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| |<math>~~~\and~~~</math>
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| |[[File:Venn 0000 1111.svg|50px]]
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| |}
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| '''[[distributivity]]:''' with various operations, especially with ''[[logical disjunction|or]]''
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| {| style="text-align: center; border: 1px solid darkgray;"
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| |-
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| |<math>~A</math>
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| |<math>\and</math>
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| |<math>(B \or C)</math>
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| | <math>\Leftrightarrow</math>
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| |<math>(A \and B)</math>
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| |<math>\or</math>
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| |<math>(A \and C)</math>
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| |-
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| |-
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| |[[File:Venn 0101 0101.svg|50px]]
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| |<math>\and</math>
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| |[[File:Venn 0011 1111.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn 0001 0101.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn 0001 0001.svg|50px]]
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| |<math>\or</math>
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| |[[File:Venn 0000 0101.svg|50px]]
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| |}
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| {| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaaaaa;"
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| ! bgcolor="#ccccff"|others
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| |-
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| with [[exclusive or]]:
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| {| style="text-align: center; border: 1px solid darkgray;"
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| |-
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| |<math>~A</math>
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| |<math>\and</math>
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| |<math>(B \oplus C)</math>
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| | <math>\Leftrightarrow</math>
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| |<math>(A \and B)</math>
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| |<math>\oplus</math>
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| |<math>(A \and C)</math>
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| |-
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| |-
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| |[[File:Venn 0101 0101.svg|50px]]
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| |<math>\and</math>
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| |[[File:Venn 0011 1100.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn 0001 0100.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn 0001 0001.svg|50px]]
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| |<math>\oplus</math>
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| |[[File:Venn 0000 0101.svg|50px]]
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| |}
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| with [[material nonimplication]]:
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| {| style="text-align: center; border: 1px solid darkgray;"
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| |-
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| |<math>~A</math>
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| |<math>\and</math>
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| |<math>(B \nrightarrow C)</math>
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| | <math>\Leftrightarrow</math>
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| |<math>(A \and B)</math>
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| |<math>\nrightarrow</math>
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| |<math>(A \and C)</math>
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| |-
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| |-
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| |[[File:Venn 0101 0101.svg|50px]]
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| |<math>\and</math>
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| |[[File:Venn 0011 0000.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn 0001 0000.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn 0001 0001.svg|50px]]
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| |<math>\nrightarrow</math>
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| |[[File:Venn 0000 0101.svg|50px]]
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| |}
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| with itself:
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| {| style="text-align: center; border: 1px solid darkgray;"
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| |-
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| |<math>~A</math>
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| |<math>\and</math>
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| |<math>(B \and C)</math>
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| | <math>\Leftrightarrow</math>
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| |<math>(A \and B)</math>
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| |<math>\and</math>
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| |<math>(A \and C)</math>
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| |-
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| |-
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| |[[File:Venn 0101 0101.svg|50px]]
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| |<math>\and</math>
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| |[[File:Venn 0000 0011.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn 0000 0001.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn 0001 0001.svg|50px]]
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| |<math>\and</math>
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| |[[File:Venn 0000 0101.svg|50px]]
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| |}
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| |}
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| '''[[idempotency]]: yes'''<br>
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| {| style="text-align: center; border: 1px solid darkgray;"
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| |-
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| |<math>~A~</math>
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| |<math>~\and~</math>
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| |<math>~A~</math>
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| | <math>\Leftrightarrow</math>
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| |<math>A~</math>
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| |-
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| |[[File:Venn01.svg|36px]]
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| |<math>~\and~</math>
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| |[[File:Venn01.svg|36px]]
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| | <math>\Leftrightarrow</math>
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| |[[File:Venn01.svg|36px]]
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| |}
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| '''[[Monotonic function#Boolean_functions|monotonicity]]: yes'''
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| {| style="text-align: center; border: 1px solid darkgray;"
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| |<math>A \rightarrow B</math>
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| | <math>\Rightarrow</math>
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| |<math>(A \and C)</math>
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| |<math>\rightarrow</math>
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| |<math>(B \and C)</math>
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| |-
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| ||[[File:Venn 1011 1011.svg|50px]]
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| | <math>\Rightarrow</math>
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| ||[[File:Venn 1111 1011.svg|50px]]
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| | <math>\Leftrightarrow</math>
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| ||[[File:Venn 0000 0101.svg|50px]]
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| |<math>\rightarrow</math>
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| ||[[File:Venn 0000 0011.svg|50px]]
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| |}
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| '''truth-preserving: yes'''<br>
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| When all inputs are true, the output is true.
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| {| style="text-align: center; border: 1px solid darkgray;"
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| |-
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| |<math>A \and B</math>
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| | <math>\Rightarrow</math>
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| |<math>A \and B</math>
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| |-
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| |[[File:Venn0001.svg|50px]]
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| | <math>\Rightarrow</math>
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| |[[File:Venn0001.svg|60px]]
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| |-
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| |<small>(to be tested)</small>
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| |}
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| '''falsehood-preserving: yes'''<br>
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| When all inputs are false, the output is false.
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| {| style="text-align: center; border: 1px solid darkgray;"
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| |-
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| |<math>A \and B</math>
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| | <math>\Rightarrow</math>
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| |<math>A \or B</math>
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| |-
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| |[[File:Venn0001.svg|60px]]
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| | <math>\Rightarrow</math>
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| |[[File:Venn0111.svg|50px]]
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| |-
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| |<small>(to be tested)</small>
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| |}
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| '''[[Hadamard transform|Walsh spectrum]]: (1,-1,-1,1)''' | |
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| '''Non[[Linear#Boolean functions|linearity]]: 1''' (the function is [[bent function|bent]])
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| If using [[binary numeral system|binary]] values for true (1) and false (0), then ''logical conjunction'' works exactly like normal arithmetic [[multiplication]].
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| ==Applications in computer engineering==
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| [[File:AND Gate diagram.svg|thumb|right|AND [[logic gate]]]]
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| In high-level computer programming and [[digital electronics]], logical conjunction is commonly represented by an infix operator, usually as a keyword such as "<code>AND</code>", an algebraic multiplication, or the ampersand symbol "<code>&</code>". Many languages also provide [[short-circuit evaluation|short-circuit]] control structures corresponding to logical conjunction.
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| Logical conjunction is often used for bitwise operations, where <code>0</code> corresponds to false and <code>1</code> to true:
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| * <code>0 AND 0</code> = <code>0</code>,
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| * <code>0 AND 1</code> = <code>0</code>,
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| * <code>1 AND 0</code> = <code>0</code>,
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| * <code>1 AND 1</code> = <code>1</code>.
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| The operation can also be applied to two binary [[words]] viewed as [[bitstring]]s of equal length, by taking the bitwise AND of each pair of bits at corresponding positions. For example:
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| * <code>11000110 AND 10100011</code> = <code>10000010</code>.
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| This can be used to select part of a bitstring using a [[Mask (computing)|bit mask]]. For example, <code>1001'''1'''101 AND 0000'''1'''000</code> = <code>0000'''1'''000</code> extracts the fifth bit of an 8-bit bitstring.
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| In [[computer networking]], bit masks are used to derive the network address of a [[subnetwork|subnet]] within an existing network from a given [[IP address]], by ANDing the IP address and the [[subnetwork#Binary subnet masks|subnet mask]].
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| Logical conjunction "<code>AND</code>" is also used in [[SQL]] operations to form [[database]] queries.
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| The [[Curry-Howard correspondence]] relates logical conjunction to [[product type]]s.
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| ==Set-theoretic correspondence==
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| The membership of an element of an [[intersection (set theory)|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]], [[commutativity]], and [[idempotence]].
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| ==Natural language==
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| The logical conjunction ''and'' in logic is related to, but not the same as, the [[grammatical conjunction]] ''and'' in natural languages.
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| English "and" has properties not captured by logical conjunction. For example, "and" sometimes implies order. 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.
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| ==See also==
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| <div style="-moz-column-count:3; column-count:3;"> | |
| * [[And-inverter graph]]
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| * [[AND gate]]
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| * [[Binary and]]
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| * [[Bitwise operation#AND|Bitwise AND]]
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| * [[Boolean algebra (logic)]]
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| * [[Boolean algebra topics]]
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| * [[Boolean conjunctive query]]
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| * [[Boolean domain]]
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| * [[Boolean function]]
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| * [[Boolean-valued function]]
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| * [[Conjunction introduction]]
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| * [[Conjunction elimination]]
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| * [[De Morgan's laws]]
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| * [[First-order logic]]
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| * [[Fréchet inequalities]]
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| * [[Grammatical conjunction]]
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| * [[Logical disjunction]]
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| * [[Logical negation]]
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| * [[Logical graph]]
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| * [[Logical value]]
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| * [[Operation (mathematics)|Operation]]
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| * [[Peano-Russell notation]]
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| * [[Propositional calculus]]
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| </div> | |
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| ==External links==
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| * {{springer|title=Conjunction|id=p/c025080}}
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| *[http://mathworld.wolfram.com/Conjunction.html Wolfram MathWorld: Conjunction]
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| == References ==
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| {{reflist}}
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| {{Logical connectives}}
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| [[Category:Logical connectives]]
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