Big-O and equals sign, abuse of notation

Question

Wikipedia says:

The statement "f(x) is O(g(x))" as defined above is usually written as 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 de Bruijn says, O(x) = O(x^2) is true but O(x^2) = O(x) is not

I understand the formal definition but not what de Bruin says. Im puzzeled by trying to understand what O(x) = O(x^2) or even O(x) is O(x^2) really means.

Intuitively I would read it as "The class of functions with complexity x is the same as the class of functions with complexity x^2". But that does not make sense.

The wikipedia talk page does not help much either.

Answer 1

In math, '=' is usually expected represent "equality", and should be an equivalence relation. This means it should be reflexive, symmetric and transitive.

As de Bruijn says, O(x) = O(x^2) is true but O(x^2) = O(x) is not

means the relation is not symmetric.

Answer 2

Intuitively I would read it as "The class of functions with complexity x is the same as the class of functions with complexity x^2". But that does not make sense.

Yes, and that is why people don't like the notation with the equals sign.

It should read as "The class of functions with complexity x is included in the class of functions with complexity x^2" or "A function with a linear upper bound for complexity is also a function with a quadratic upper complexity bound" (where of course the quadratic bound is not very tight).