Editing Exclusive or (section)

==Properties==
{{glossary}}
{{term|[[Commutative property|Commutativity]]: yes}}{{defn|
{{aligned table
 | style=text-align:center; border:1px solid darkgrey;
 | rowstyle=;
 | cols=3
 | <math>A \oplus B</math>
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | <math>B \oplus A</math>
 | [[File:Venn0110.svg|50px]]
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | [[File:Venn0110.svg|50px]]
}}
}}
{{term|[[Associative property|Associativity]]: yes}}{{defn|
{{aligned table
 | style=text-align:center; border:1px solid darkgrey;
 | rowstyle=;
 | cols=9
 | <math>~A</math>
 | <math>~~~\oplus~~~</math>
 | <math>(B \oplus C)</math>
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 |
 |
 | <math>(A \oplus B)</math>
 | <math>~~~\oplus~~~</math>
 | <math>~C</math>
 | [[File:Venn 0101 0101.svg|50px]]
 | <math>~~~\oplus~~~</math>
 | [[File:Venn 0011 1100.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 0110 0110.svg|50px]]
 | <math>~~~\oplus~~~</math>
 | [[File:Venn 0000 1111.svg|50px]]
 }}
}}
{{term|[[Distributive property|Distributivity]]:}}{{defn|The exclusive or does not distribute over any binary function (not even itself), but [[Logical conjunction#Properties|logical conjunction distributes over exclusive or]]. <math>C\land(A \oplus B )= (C\land A) \oplus (C\land B)</math> (Conjunction and exclusive or form the multiplication and addition operations of a [[Field (mathematics)|field]] [[GF(2)]], and as in any field they obey the distributive law.)}}
{{term|[[Idempotence|Idempotency]]: no}}{{defn|
 {{aligned table
 | style=text-align:center; border:1px solid darkgrey;
 | rowstyle=;
 | cols=7
 | <math>~A~</math> 
 | <math>~\oplus~</math>
 | <math>~A~</math>
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | <math>~0~</math>
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\nLeftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | <math>~A~</math>
 | [[File:Venn01.svg|36px]]
 | <math>~\oplus~</math>
 | [[File:Venn01.svg|36px]]
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | [[File:Venn00.svg|36px]]
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\nLeftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | [[File:Venn01.svg|36px]]
 }}
}}
{{term|[[Monotone Boolean function|Monotonicity]]: no}}{{defn|
 {{aligned table
 | style=text-align:center; border:1px solid darkgrey;
 | rowstyle=;
 | cols=7
 | <math>A \rightarrow B</math>
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 |
 |
 | <math>(A \oplus C)</math>
 | <math>\rightarrow</math>
 | <math>(B \oplus C)</math>
 | [[File:Venn 1011 1011.svg|50px]]
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | [[File:Venn 1011 1101.svg|50px]]
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | [[File:Venn 0101 1010.svg|50px]]
 | <math>\rightarrow</math>
 | [[File:Venn 0011 1100.svg|50px]]
 }}
}}
{{term|Truth-preserving: no}}{{defn|When all inputs are true, the output is not true.
 {{aligned table
 | style=text-align:center; border:1px solid darkgrey;
 | rowstyle=;
 | cols=3
 | <math>A \land B</math>
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | <math>A \oplus B</math>
 | [[File:Venn0001.svg|50px]]
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\nRightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | [[File:Venn0110.svg|60px]]
 }}
}}
{{term|Falsehood-preserving: yes}}{{defn|When all inputs are false, the output is false.
 {{aligned table
 | style=text-align:center; border:1px solid darkgrey;
 | rowstyle=;
 | cols=3
 | <math>A \oplus B</math>
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | <math>A \lor B</math>
 | [[File:Venn0110.svg|60px]]
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | [[File:Venn0111.svg|50px]]
 }}
}}
{{term|[[Hadamard transform|Walsh spectrum]]: (2,0,0,−2)}}
{{term|Non-[[Linear#Boolean functions|linearity]]: 0}}{{defn|The function is linear.}}
{{term|Involution:}}{{defn|Exclusive or with one specified input, as a function of the other input, is an [[Involution (mathematics)|involution]] or self-inverse function; applying it twice leaves the variable input unchanged.
 {{aligned table
 | style=text-align:center; border:1px solid darkgrey;
 | rowstyle=;
 | cols=5
 | <math>~A\oplus B~</math> 
 | <math>~\oplus~</math>
 | <math>~B~</math>
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | <math>~A~</math>
 | [[File:Venn0110.svg|50px]]
 | <math>~\oplus~</math>
 | [[File:Venn0011.svg|50px]]
 | &nbsp;&nbsp;&nbsp;&nbsp;<math>\Leftrightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;
 | [[File:Venn0101.svg|50px]]
 }}
}}
{{glossary end}}

If using [[binary numeral system|binary]] values for true (1) and false (0), then ''exclusive or'' works exactly like [[addition]] [[Modular arithmetic|modulo]] 2.