Editing Big O notation (section)

=== Other notation ===
In their book ''[[Introduction to Algorithms]]'', [[Thomas H. Cormen|Cormen]], [[Charles E. Leiserson|Leiserson]], [[Ronald L. Rivest|Rivest]] and [[Clifford Stein|Stein]] consider the set of functions ''f'' which satisfy

:<math> f(n) = O(g(n))\quad(n\to\infty)~.</math>

In a correct notation this set can, for instance, be called ''O''(''g''), where

<math display=block>O(g) = \{ f : \text{there exist positive constants}~c~\text{and}~n_0~\text{such that}~0 \le f(n) \le c g(n) \text{ for all } n \ge n_0 \}.</math><ref>{{cite book |  isbn=978-0-262-53305-8 |author1=Cormen, Thomas H. |author2=Leiserson, Charles E. |author3=Rivest, Ronald L. |title=Introduction to Algorithms |location=Cambridge/MA |publisher=MIT Press |edition=3rd |year=2009 |page=47 |quote=When we have only an asymptotic upper bound, we use O-notation. For a given function ''g''(''n''), we denote by ''O''(''g''(''n'')) (pronounced "big-oh of ''g'' of ''n''" or sometimes just "oh of ''g'' of ''n''") the set of functions ''O''(''g''(''n'')) = { ''f''(''n'') : there exist positive constants ''c'' and ''n''<sub>0</sub> such that 0 ≤ ''f''(''n'') ≤ ''cg''(''n'') for all ''n'' ≥ ''n''<sub>0</sub>} }}</ref>

The authors state that the use of equality operator (=) to denote set membership rather than the set membership operator (∈) is an abuse of notation, but that doing so has advantages.<ref name="clrs3">{{cite book |isbn=978-0-262-53305-8 |author1=Cormen, Thomas H. |author2=Leiserson, Charles E. |author3=Rivest, Ronald L. |title=Introduction to Algorithms |url=https://archive.org/details/introductiontoal00corm_805 |url-access=limited |location=Cambridge/MA |publisher=MIT Press |edition=3rd |year=2009 |page=[https://archive.org/details/introductiontoal00corm_805/page/n65 45] |quote=Because ''θ''(''g''(''n'')) is a set, we could write "''f''(''n'') ∈ ''θ''(''g''(''n''))" to indicate that ''f''(''n'') is a member of ''θ''(''g''(''n'')). Instead, we will usually write ''f''(''n'') = ''θ''(''g''(''n'')) to express the same notion. You might be confused because we abuse equality in this way, but we shall see later in this section that doing so has its advantages.}}</ref> Inside an equation or inequality, the use of asymptotic notation stands for an anonymous function in the set ''O''(''g''), which eliminates lower-order terms, and helps to reduce inessential clutter in equations, for example:<ref>{{cite book |isbn=978-0-262-53305-8 |author1=Cormen, Thomas H. |author2=Leiserson, Charles E. |author3=Rivest, Ronald L. |title=Introduction to Algorithms |url=https://archive.org/details/introductiontoal00corm_805 |url-access=limited |location=Cambridge/MA |publisher=MIT Press |edition=3rd |year=2009 |page=[https://archive.org/details/introductiontoal00corm_805/page/n69 49] |quote=When the asymptotic notation stands alone (that is, not within a larger formula) on the right-hand side of an equation (or inequality), as in n = O(n<sup>2</sup>), we have already defined the equal sign to mean set membership: n ∈ O(n<sup>2</sup>). In general, however, when asymptotic notation appears in a formula, we interpret it as standing for some anonymous function that we do not care to name. For example, the formula 2''n''<sup>2</sup> + 3''n'' + 1 = 2''n''<sup>2</sup> + ''θ''(''n'') means that 2''n''<sup>2</sup> + 3''n'' + 1 = 2''n''<sup>2</sup> + ''f''(''n''), where ''f''(''n'') is some function in the set ''θ''(''n''). In this case, we let ''f''(''n'') = 3''n'' + 1, which is indeed in ''θ''(''n''). Using asymptotic notation in this manner can help eliminate inessential detail and clutter in an equation.}}</ref>

:<math> 2n^2 + 3n + 1=2n^2 + O(n).</math>