Editing Logical biconditional (section)

==Properties==
'''[[Commutativity]]: Yes'''
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>A \leftrightarrow B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>B \leftrightarrow A</math>
|-
|[[File:Venn1001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn1001.svg|50px]]
|}

'''[[Associativity]]: Yes'''
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>~A</math>
|<math>~~~\leftrightarrow~~~</math>
|<math>(B \leftrightarrow C)</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|
|
|<math>(A \leftrightarrow B)</math>
|<math>~~~\leftrightarrow~~~</math>
|<math>~C</math>
|-
|[[File:Venn 0101 0101.svg|50px]]
|<math>~~~\leftrightarrow~~~</math>
|[[File:Venn 1100 0011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0110 1001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 1001 1001.svg|50px]]
|<math>~~~\leftrightarrow~~~</math>
|[[File:Venn 0000 1111.svg|50px]]
|}

'''[[Distributivity]]:'''  Biconditional doesn't distribute over any binary function (not even itself), but [[Logical disjunction#Properties|logical disjunction distributes]] over biconditional.

'''[[Idempotency]]: No'''<br />
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>~A~</math>  
|<math>~\leftrightarrow~</math> 
|<math>~A~</math> 
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>~1~</math> 
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nLeftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>~A~</math> 
|-
|[[File:Venn01.svg|36px]] 
|<math>~\leftrightarrow~</math> 
|[[File:Venn01.svg|36px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn11.svg|36px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nLeftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn01.svg|36px]]
|}

'''[[Monotone Boolean function|Monotonicity]]: No'''
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>A \rightarrow B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|
|
|<math>(A \leftrightarrow C)</math>
|<math>\rightarrow</math>
|<math>(B \leftrightarrow C)</math>
|-
||[[File:Venn 1011 1011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
||[[File:Venn 1101 1011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
||[[File:Venn 1010 0101.svg|50px]]
|<math>\rightarrow</math>
||[[File:Venn 1100 0011.svg|50px]]
|}

'''Truth-preserving: Yes'''<br />
When all inputs are true, the output is true.
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>A \land B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>A \leftrightarrow B</math>
|-
|[[File:Venn0001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn1001.svg|60px]]
|}

'''Falsehood-preserving: No'''<br />
When all inputs are false, the output is not false.
{| style="text-align:center; border:1px solid darkgrey;"
|-
|<math>A \leftrightarrow B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>A \lor B</math>
|-
|[[File:Venn1001.svg|60px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn0111.svg|50px]]
|}

'''[[Hadamard transform|Walsh spectrum]]: (2,0,0,2)'''

'''Non[[Linear#Boolean functions|linearity]]: 0''' (the function is linear)