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Assuming that $g$ is eventually positive, then

$f = O(g)$ means that $|f(x)| \le cg(x)$ for some positive constant $c$ and all sufficiently large $x$;

$f = \Omega(g)$ means that $ f(x) \ge cg(x)$ for some positive constant $c$ and all sufficiently large $x$.

Why couldn't $|f(x)|$ be also used in the definition for $\Omega$ ? Why is there this difference?

Robert Z
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chen
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  • I think that $g(x)>0$ is assumed in both definitions. Then $f(x)>0$ as well in the $\Omega$ notation. – Olivier Moschetta Dec 09 '16 at 10:11
  • O notation can be extended by redefining $f = O(g)$ as $|f| \leq c|g|$ and $f = \Omega(g)$ as $|f| \geq c|g|$. It just depends on which definitions you're using. – Antonio Vargas Dec 10 '16 at 20:53

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