Unfortunately, the way we write big oh notation can be misleading to the naıve reader. This section presents two fallacies which arise because of a misinterpretation of the notation.

FallacyGiven that and , then .

Consider the equations:

Clearly, it is reasonable to conclude that .

However, consider these equations:

It *does not* follow that .
E.g., and are both , but they are not equal.

FallacyIff(n)=O(g(n)), then .

Consider functions *f*, *g*, and *h*, such that *f*(*n*)=*h*(*g*(*n*)).
It is reasonable to conclude that
provided that is an invertible function.
However, while we may write *f*(*n*)=*O*(*h*(*n*)),
the equation is nonsensical and meaningless.
Big oh is not a mathematical function,
so it has no inverse!

The reason for these difficulties is that
we should read the notation as
``*f*(*n*) is big oh *n* squared''
not
``*f*(*n*) equals big oh of *n* squared.''
The equal sign in the expression does not really denote mathematical equality!
And the use of the functional form, ,
does not really mean that *O* is a mathematical function!

Copyright © 1997 by Bruno R. Preiss, P.Eng. All rights reserved.