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Logical biconditional
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===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 ---> |}
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