I’m having trouble understanding what constitutes a “minimal sum section” of an integer array. My book defines it as the following:

Let $ a[0],\dots, a[n-1]$ be the integer values of an array $ a$ .

Asectionof $ a$ is a continuous piece $ a[i],\dots,a[j]$ , where $ 0\le i \le j < n$ . We write $ S_i,_j$ for the sum of that section: $ a[i] + a[i+1]+\dots+a[j]$ .

Aminimal-sum sectionis a section $ a[i],\dots,a[j]$ of $ a$ such that the sum $ S_{i,j}$ is less than or equal to the sum $ S_{i’,j’}$ of any other section $ a[i’],\dots,a[j’]$ of $ a$ .

My confusion comes with one of the examples that follow this definition:

The array [1,-1,3,-1,1] has two minimal-sum sections [1,-1] and [-1,1] with minimal sum 0.

But, wouldn’t the minimal sum section be $ [-1]$ ?

In a later example they give:

array $ [-1,3,-2]$

Minimal sum

section $ [-2]$

So, in the last example they definitely counted one element as the minimal sum section, but not in the first one. Any clarification on why this is so would be greatly appreciated.