Editing Logical biconditional (section)

==Definition==

[[Logical equality]] (also known as biconditional) 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 both operands are false or both operands are true.<ref name=":1" />

===Truth table===
The following is a truth table for <math>A \leftrightarrow B</math>:

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

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When more than two statements are involved, combining them with <math>\leftrightarrow</math> might be ambiguous. For example, the statement

:<math>x_1 \leftrightarrow x_2 \leftrightarrow x_3 \leftrightarrow \cdots \leftrightarrow x_n</math>

may be interpreted as

:<math>(((x_1 \leftrightarrow x_2) \leftrightarrow x_3) \leftrightarrow \cdots) \leftrightarrow x_n</math>,

or may be interpreted as saying that all {{math|''x<sub>i</sub>''}} are ''jointly true or jointly false'':
:<math>(x_1 \land \cdots \land x_n) \lor (\neg x_1 \land \cdots \land \neg x_n)</math>

As it turns out, these two statements are only the same when zero or two arguments are involved. In fact, the following truth tables only show the same bit pattern in the line with no argument and in the lines with two arguments:

[[File:Variadic logical XAND.svg|thumb|left|220px|<math>~x_1 \leftrightarrow \cdots \leftrightarrow x_n</math><br />meant as equivalent to<br /><math>\neg~(\neg x_1 \oplus \cdots \oplus \neg x_n)</math><br /><br />The central Venn diagram below,<br />and line ''(ABC&nbsp;&nbsp;)'' in this matrix<br />represent the same operation.]]

[[File:Variadic logical all or nothing.svg|thumb|right|220px|<math>~x_1 \leftrightarrow \cdots \leftrightarrow x_n</math><br />meant as shorthand for<br /><math>(~x_1 \land \cdots \land x_n~)</math><br /><math>\lor~(\neg x_1 \land \cdots \land \neg x_n)</math><br /><br />The Venn diagram directly below,<br />and line ''(ABC&nbsp;&nbsp;)'' in this matrix<br />represent the same operation.]]
{{-}}
The left Venn diagram below, and the lines ''(AB&nbsp;&nbsp;&nbsp;&nbsp;)'' in these matrices represent the same operation.

===Venn diagrams===
Red areas stand for true (as in [[Image:Venn0001.svg|40px]] for ''[[Logical conjunction|and]]'').

{| border="0" style="width:100%"
| style="vertical-align:top;"|<!--- START LEFT TABLE IN TABLE --->
{|  style="background:#f9f9f9; border:1px solid #ccc; float:left;"
|-
| [[Image:Venn1001.svg|220px]]
|-
| The biconditional of two statements<br />is the [[negation]] of the [[exclusive or]]:
|- style="text-align:center;"
|<math>~A \leftrightarrow B~~\Leftrightarrow~~\neg(A \oplus B)</math>

[[File:Venn1001.svg|40px]] <math>\Leftrightarrow \neg</math> [[Image:Venn0110.svg|40px]]
|}<!--- END LEFT TABLE IN TABLE --->
| style="width: 100px"|
| style="vertical-align:top;"|<!--- START CENTRAL TABLE IN TABLE --->
{|  style="background:#f9f9f9; border:1px solid #ccc; margin:auto;"
|-
| [[File:Venn 0110 1001.svg|220px]]
|-
| The biconditional and the<br />exclusive or of three statements<br />give the same result:<br />
<math>~A \leftrightarrow B \leftrightarrow C~~\Leftrightarrow</math><br />
<math>~A \oplus B \oplus C</math>

[[File:Venn 1001 1001.svg|40px]] <math>\leftrightarrow</math> [[File:Venn 0000 1111.svg|40px]]
<math>~~\Leftrightarrow~~</math>

[[File:Venn 0110 0110.svg|40px]] <math>\oplus</math> [[File:Venn 0000 1111.svg|40px]]
<math>~~\Leftrightarrow~~</math> [[File:Venn 0110 1001.svg|40px]]
|}<!--- END CENTRAL TABLE IN TABLE --->
| style="width: 100px"|
| style="vertical-align:top;"|<!--- START RIGHT TABLE IN TABLE --->
{|  style="background:#f9f9f9; border:1px solid #ccc; float:right;"
|-
| [[File:Venn 1000 0001.svg|220px]]
|-
| But <math>~A \leftrightarrow B \leftrightarrow C</math><br />may also be used as an abbreviation<br />for <math>(A \leftrightarrow B) \land (B \leftrightarrow C)</math>

[[File:Venn 1001 1001.svg|40px]] <math>\land</math> [[File:Venn 1100 0011.svg|40px]]
<math>~~\Leftrightarrow~~</math> [[File:Venn 1000 0001.svg|40px]]
|}<!--- END RIGHT TABLE IN TABLE --->
|}