Editing Logical conjunction (section)

==Definition==
In [[classical logic]], '''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'' [[if and only if]] (also known as iff) both of its operands are true.<ref name=":1" /><ref name=":2" />

The conjunctive [[identity element|identity]] 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]], the empty conjunction (AND-ing over an empty set of operands) is often defined as having the result true.

===Truth table===
[[File:Variadic logical AND.svg|thumb|Conjunctions of the arguments on the left — The [[truth value|true]] [[bit]]<nowiki>s</nowiki> form a [[Sierpinski triangle]].]]

The [[truth table]] of <math>A \land B</math>:<ref name=":2" /><ref name=":1" />

{{2-ary truth table|0|0|0|1|<math>A \land B</math>}}

===Defined by other operators===
In systems where logical conjunction is not a primitive, it may be defined as<ref>{{Cite web |url=http://www.logicmatters.net/resources/pdfs/ProofSystems.pdf |title=Types of proof system |last=Smith |first=Peter |page=4}}</ref>
:<math>A \land B = \neg(A \to \neg B) </math>
It can be checked by the following truth table (compare the last two columns):

{{2-ary truth table
|1|0|1|0|<math>\neg B</math>
|thick
|1|1|1|0|<math>A \rightarrow \neg B</math>
|
|0|0|0|1|<math>\neg(A \rightarrow \neg B)</math>
|
|0|0|0|1|<math>A \land B</math>
}}
or
:<math>A \land B = \neg(\neg A \lor \neg B).</math>
It can be checked by the following truth table (compare the last two columns):

{{2-ary truth table
|1|1|0|0|<math>\neg A</math>
|
|1|0|1|0|<math>\neg B</math>
|thick
|1|1|1|0|<math>\neg A \or \neg B</math>
|
|0|0|0|1|<math>\neg(\neg A \or \neg B)</math>
|
|0|0|0|1|<math>A \land B</math>
}}