Editing Big O notation (section)

=== Equals sign ===
The statement "{{math|''f''(''x'')}} is {{math|''O''[''g''(''x'')]}}" as defined above is usually written as {{math|1= ''f''(''x'') {{=}} ''O''[''g''(''x'')]}}. Some consider this to be an [[abuse of notation]], since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As [[Nicolaas Govert de Bruijn|de Bruijn]] says, {{math|1= ''O''[''x''] {{=}} ''O''[''x''<sup>2</sup>]}} is true but {{math|1= ''O''[''x''<sup>2</sup>] {{=}} ''O''[''x'']}} is not.<ref name=deBruijn>{{cite book |first=N.G. |last=de Bruijn | author-link=N. G. de Bruijn | title=Asymptotic Methods in Analysis | place=Amsterdam | publisher=North-Holland | year=1958 | pages=5–7 | url=https://books.google.com/books?id=_tnwmvHmVwMC&q=%22The+trouble+is%22&pg=PA5 | isbn=978-0-486-64221-5 | access-date=2021-09-15 | archive-date=2023-01-17 | archive-url=https://web.archive.org/web/20230117051949/https://books.google.com/books?id=_tnwmvHmVwMC&q=%22The+trouble+is%22&pg=PA5 | url-status=live }}</ref> [[Donald Knuth|Knuth]] describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like {{math|1= ''n'' {{=}} ''n''<sup>2</sup>}} from the identities {{math|1= ''n'' {{=}} ''O''[''n''<sup>2</sup>]}} and {{math|1= ''n''<sup>2</sup> {{=}} ''O''[''n''<sup>2</sup>]}}".<ref name="Concrete Mathematics">{{cite book |last1=Graham |first1=Ronald |author1-link=Ronald Graham |first2=Donald |last2=Knuth |author2-link=Donald Knuth |last3=Patashnik |first3=Oren |author3-link=Oren Patashnik |title=Concrete Mathematics |location=Reading, Massachusetts |publisher=Addison–Wesley |edition=2 |date=1994 |page=446 |url=https://books.google.com/books?id=pntQAAAAMAAJ |isbn=978-0-201-55802-9 |access-date=2016-09-23 |archive-date=2023-01-17 |archive-url=https://web.archive.org/web/20230117051955/https://books.google.com/books?id=pntQAAAAMAAJ |url-status=live }}</ref> In another letter, Knuth also pointed out that<ref>{{cite journal | author=Donald Knuth | title=Teach Calculus with Big O | date=June–July 1998 | journal=[[Notices of the American Mathematical Society]] | volume=45 | issue=6 | page=687 | url=https://www.ams.org/notices/199806/commentary.pdf | access-date=2021-09-05 | archive-date=2021-10-14 | archive-url=https://web.archive.org/web/20211014070416/https://www.ams.org/notices/199806/commentary.pdf | url-status=live }} ([http://www-cs-staff.stanford.edu/~knuth/ocalc.tex Unabridged version] {{Webarchive|url=https://web.archive.org/web/20080513234708/http://www-cs-staff.stanford.edu/~knuth/ocalc.tex |date=2008-05-13 }})</ref>
{{blockquote|text= the equality sign is not symmetric with respect to such notations [as, in this notation,] mathematicians customarily use the '=' sign as they use the word 'is' in English: Aristotle is a man, but a man isn't necessarily Aristotle.}}

For these reasons, it would be more precise to use [[set notation]] and write {{math|''f''(''x'') ∈ ''O''[''g''(''x'')]}}{{snd}} read as: "{{math|''f''(''x'')}} [[Element (mathematics)#Notation and terminology|is an element of]] {{math|''O''[''g''(''x'')]}}", or "{{math|''f''(''x'')}} is in the set {{math|''O''[''g''(''x'')]}}"{{snd}} thinking of {{math|''O''[''g''(''x'')]}} as the class of all functions {{math|''h''(''x'')}} such that {{math|{{abs|''h''(''x'')}} ≤ ''C'' {{abs|''g''(''x'')}}}} for some positive real number {{mvar|C}}.<ref name="Concrete Mathematics" /> However, the use of the equals sign is customary.<ref name=deBruijn/><!-- p. 7: "Once this warning has been given, there is, however, not much harm in using the sign =, and we shall maintain it, for no other reason than that it is customary." --><ref name="Concrete Mathematics"/><!-- p. 446: "why don't we use ‘⊆’ instead of abusing the equals sign? There are four reasons. First, tradition ..." -->