Difference between O(m+n) and O(mn)?

Question

I was trying to find the complexities of an algorithm via different approaches. Mathematically I came across one O(m+n) and another O(mn) approach. However I am unable to gr

Answer 1

O(m+n) example:

for(int i = 0, i < m, i++)
    //code
for(int j = 0, j < n, j++)
    //code

m iterations of code happen. Then n iterations of code happens.

O(mn) example:

for(int i = 0, i < m, i++)
    for(int j = 0, j < n, j++)
        //code

For every iteration of m, we have n iterations of code. Imagine iterating over a non-square 2D array.

m and n do not necessarily equal the same value. If they did equal the same value, then for O(m+n):

O(m+n) => O(m+m) => O(2m) => O(m)

I'd recommend looking at this question/answer in order to understand that last transition.

And for O(mn):

O(mn) => O(mm) => O(m^2)