big-o - 算法统治

Question

为考试而学习并得到这个问题:

Comparing two algorithms with asymptotic complexities O(n) and O(n + log(n)), 
which one of the following is true?

A) O(n + log(n)) dominates O(n)
B) O(n) dominates O(n + log(n))
C) Neither algorithm dominates the other.

O(n) 支配 log(n) 正确吗？那么在这种情况下，我们是否只从两者中取 o(n) 并推断出两者都不占主导地位？

Answer 1

[C] 是真的，因为 the summation property Big-O

Summation O(f(n)) + O(g(n)) -> O(max(f(n), g(n))) For example: O(n^2) + O(n) = O(n^2)

In Big-O, you only care about the largest-growing function and ignore all the other additives.

Edit: originally I put [A] as an answer, I just didn't put much attention to all the options and misinterpreted the [A] option. Here is more formal proof

O(n) ~ O(n + log(n))      <=>
O(n) ~ O(n) + O(log(n))   <=>
O(n) ~ O(n).