Editing Big O notation (section)

=== Multiple uses ===
In more complicated usage, {{math|''O''(·)}} can appear in different places in an equation, even several times on each side. For example, the following are true for <math>n\to\infty</math>:
<math display="block">
\begin{align}
(n+1)^2 & = n^2 + O(n), \\
(n + O(n^{1/2})) \cdot (n + O(\log n))^2 & = n^3 + O(n^{5/2}), \\
n^{O(1)} & = O(e^n).
\end{align}</math>
The meaning of such statements is as follows: for {{em|any}} functions which satisfy each {{math|''O''(·)}} on the left side, there are {{em|some}} functions satisfying each {{math|''O''(·)}} on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function {{math|1= ''f''(''n'') = ''O''(1)}}, there is some function {{math|1= ''g''(''n'') = ''O''(''e''<sup>''n''</sup>)}} such that {{math|1= ''n''<sup>''f''(''n'')</sup> = ''g''(''n'')}}". In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side. In this use the "{{math|1= =}}" is a formal symbol that unlike the usual use of "{{math|1= =}}" is not a [[symmetric relation]]. Thus for example {{math|1=''n''<sup>''O''(1)</sup> = ''O''(''e''<sup>''n''</sup>)}} does not imply the false statement {{math|1=''O''(''e''<sup>''n''</sup>) = ''n''<sup>''O''(1)</sup>}}.