Combining Predicate Logic and BigO 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.

• 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? Nov 6 '19 at 7:55

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 $$f$$.