Editing Commutative property (section)

== Definition ==

A [[binary operation]] <math>*</math> on a [[Set (mathematics)|set]] ''S'' is ''commutative'' if
<math display="block">x * y = y * x </math>
for all <math> x,y \in S</math>.{{sfn|Saracino|2008|p=[https://books.google.com/books?id=GW4fAAAAQBAJ&pg=PA11 11]}} An operation that is not commutative is said to be ''noncommutative''.{{sfn|Hall|1966|pp=[https://books.google.com/books?id=qqs8AAAAIAAJ&pg=PA262 262–263]}}

One says that {{mvar|x}} ''commutes'' with {{math|''y''}} or that {{mvar|x}} and {{mvar|y}} ''commute'' under <math>*</math> if{{sfn|Lovett|2022|p=[http://books.google.com/books?id=vp0IEQAAQBAJ&pg=PA12 12]}}
<math display="block"> x * y = y * x.</math>

So, an operation is commutative if every two elements commute.{{sfn|Lovett|2022|p=[http://books.google.com/books?id=vp0IEQAAQBAJ&pg=PA12 12]}} An operation is noncommutative if there are two elements such that <math> x * y \ne y * x.</math> This does not exclude the possibility that some pairs of elements commute.{{sfn|Hall|1966|pp=[https://books.google.com/books?id=qqs8AAAAIAAJ&pg=PA262 262–263]}}