Editing Logical conjunction (section)

==Properties==
'''[[Commutative property|commutativity]]: yes'''
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>A \land B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>B \land A</math>
|-y
|[[File:Venn0001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn0001.svg|50px]]
|}

'''[[associativity]]: yes<ref name=":13">{{Cite book |last=Howson |first=Colin |title=Logic with trees: an introduction to symbolic logic |date=1997 |publisher=Routledge |isbn=978-0-415-13342-5 |location=London; New York |pages=38}}</ref>'''
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>~A</math>
|<math>~~~\land~~~</math>
|<math>(B \land C)</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|
|
|<math>(A \land B)</math>
|<math>~~~\land~~~</math>
|<math>~C</math>
|-
|[[File:Venn 0101 0101.svg|50px]]
|<math>~~~\land~~~</math>
|[[File:Venn 0000 0011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0000 0001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0001 0001.svg|50px]]
|<math>~~~\land~~~</math>
|[[File:Venn 0000 1111.svg|50px]]
|}

'''[[distributivity]]:''' with various operations, especially with ''[[logical disjunction|or]]''
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>~A</math>
|<math>\land</math>
|<math>(B \lor C)</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|
|
|<math>(A \land B)</math>
|<math>\lor</math>
|<math>(A \land C)</math>
|-
|-
|[[File:Venn 0101 0101.svg|50px]]
|<math>\land</math>
|[[File:Venn 0011 1111.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0001 0101.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0001 0001.svg|50px]]
|<math>\lor</math>
|[[File:Venn 0000 0101.svg|50px]]
|}

{| class="collapsible collapsed" style="width: 100%; border: 1px solid #aaaaaa;"
! bgcolor="#ccccff"|others
|-
| 
with [[exclusive or]]:
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>~A</math>
|<math>\land</math>
|<math>(B \oplus C)</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|
|
|<math>(A \land B)</math>
|<math>\oplus</math>
|<math>(A \land C)</math>
|-
|-
|[[File:Venn 0101 0101.svg|50px]]
|<math>\land</math>
|[[File:Venn 0011 1100.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0001 0100.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0001 0001.svg|50px]]
|<math>\oplus</math>
|[[File:Venn 0000 0101.svg|50px]]
|}

with [[material nonimplication]]:
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>~A</math>
|<math>\land</math>
|<math>(B \nrightarrow C)</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|
|
|<math>(A \land B)</math>
|<math>\nrightarrow</math>
|<math>(A \land C)</math>
|-
|-
|[[File:Venn 0101 0101.svg|50px]]
|<math>\land</math>
|[[File:Venn 0011 0000.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0001 0000.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0001 0001.svg|50px]]
|<math>\nrightarrow</math>
|[[File:Venn 0000 0101.svg|50px]]
|}

with itself:
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>~A</math>
|<math>\land</math>
|<math>(B \land C)</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|
|
|<math>(A \land B)</math>
|<math>\land</math>
|<math>(A \land C)</math>
|-
|-
|[[File:Venn 0101 0101.svg|50px]]
|<math>\land</math>
|[[File:Venn 0000 0011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0000 0001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn 0001 0001.svg|50px]]
|<math>\land</math>
|[[File:Venn 0000 0101.svg|50px]]
|}
|}

'''[[idempotency]]: yes'''<br>
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>~A~</math>
|<math>~\land~</math>
|<math>~A~</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>A~</math>
|-
|[[File:Venn01.svg|36px]]
|<math>~\land~</math>
|[[File:Venn01.svg|36px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn01.svg|36px]]
|}

'''[[Monotonic function#In Boolean functions|monotonicity]]: yes'''
{| style="text-align: center; border: 1px solid darkgray;"
|<math>A \rightarrow B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|
|
|<math>(A \land C)</math>
|<math>\rightarrow</math>
|<math>(B \land C)</math>
|-
||[[File:Venn 1011 1011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
||[[File:Venn 1111 1011.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
||[[File:Venn 0000 0101.svg|50px]]
|<math>\rightarrow</math>
||[[File:Venn 0000 0011.svg|50px]]
|}

'''truth-preserving: yes'''<br>When all inputs are true, the output is true.
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>A \land B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>A \land B</math>
|-
|[[File:Venn0001.svg|50px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn0001.svg|60px]]
|-
|
|
|{{small|(to be tested)}}
|}

'''falsehood-preserving: yes'''<br>When all inputs are false, the output is false.
{| style="text-align: center; border: 1px solid darkgray;"
|-
|<math>A \land B</math>
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|<math>A \lor B</math>
|-
|[[File:Venn0001.svg|60px]]
|&nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
|[[File:Venn0111.svg|50px]]
|-
|{{small|(to be tested)}}
|
|
|}

'''[[Hadamard transform|Walsh spectrum]]: (1,-1,-1,1)'''

'''Non[[Linear#Boolean functions|linearity]]: 1''' (the function is [[bent function|bent]])

If using [[binary numeral system|binary]] values for true (1) and false (0), then ''logical conjunction'' works exactly like normal arithmetic [[multiplication]].