I am a beginner to predicate logic and BigO and am having though time understanding the definition of BigO in terms of predicate logic in the picture attached. I particularly am unable to understand what is meant by n0 in this context. Any help is greatly appreciated.

  • $\begingroup$ I don't think that this question is about first-order logic at all, he definition just uses the common symbols $\forall, \exists$. Have you checked common resources what the Landau-$\mathcal{O}$ notation means? $\endgroup$
    – ttnick
    Nov 6, 2019 at 7:55

1 Answer 1


What the definition is saying is: $f\in O(g)$ if there's a number $n_0$ such that $f(n)$ is less than some multiple of $g(n)$ (that's where the $c$ comes in) for all $n$ larger than $n_0$. In simple terms, it's saying that $f$ is eventually bounded by some fixed multiple of $g$.

Big-O notation is essentially a simplification describing the ultimate rate of growth of a function. Here we don't care at all if, say, $f(n)$ is bigger than $43\cdot g(n)$ for the first thousand values of $n$: if $f(n)<43\cdot g(n)$ for all the rest (the bigger values of $n$), we've established that a multiple of $g$ is an upper bound on the values of $f$.

  • $\begingroup$ Thanks a lot! Highly appreciated. $\endgroup$
    – Dhruv
    Nov 6, 2019 at 1:32

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