Is or is not A set determined by its ancestors (elements)? this post says

Axiom 1a. A set is determined by its elements 
Remark 1. It is
  important to notice that this axiom is a non trivial assertion about
  belonging. To understand this consider an analogous situation in which
  we consider human beings in the place of sets and elements, and x ∈ A
  means x is an ancestor of A. Then clearly A is not determined by its
  ancestors.

I consider ancestors as elements here, which I am not sure I understand it the right way, if yes, "A is not determined by its ancestors" seems to be a typo, which should be "A is determined by its ancestors", is it?
let A = {'a', 'b', 'c'}
is 'a' an ancestor and an element of set A? I suppose it is.
are 'a', 'b', 'c' ancestors and elements of set A? I suppose they all are.
biologically, this makes definitely sense.

My sister and I have exactly the same ancestors but we are different
  people

mathematically, I cannot understand 'a', 'b', 'c' are ancestors of set A while set A is not determined the ancestors {'a', 'b', 'c'}.
 A: No, you are not determined by your ancestor, since you could have siblings. If I have a brother, then we have exactly the same ancestors.
You are not determined by your children either, since plenty of people in history did not have any children.

To your edit:


*

*Do not take analogies too seriously. Stick to the definitions instead. Foregoing this idea of ancestors and people and getting back to sets and elements instead.

*Note that there is no typo. So ancestors do not determine a person, whereas elements do determine a set.
A: The phrase

$A$ is determined by its elements

means that if you know all the elements of $A$ you know what the set $A$ is. There can't be two different sets that contain the precisely the same elements.
The phrase

$A$ is determined by its ancestors

is false. My sister and I have exactly the same ancestors but we are different people.
A: You are missing the entire point.  The point is that a Set is NOT a person, and a person's ancestors are NOT its elements.
A person is not determined by its ancestors.  My sister and I have the exact same ancestors.  But we are not the same person.
Now you are arguing that if we called a set a "person" (and if we called a set an "elephant") and if we called the elements of a set its "ancestors" (or "pink go-go dancers") then we would say "a person is determined by its ancestor" and "an elephant is determined by its pink go-go dancers".
But that is not what the author is trying to do at all.  The author is trying to point out that the true statement "A set is determined by by its elements" is not as trivial as it sounds.  It sounds trivial.  The elements belong to a set; a set was made by it's elements. And if a set had different elements it would be a different set. So it sounds trivial to say "a set is defined entirely by what elements it has".
But the author is trying to show it is not trivial by showing a case where it wouldn't be true.  Our ancestors belong to us (we all have them); and we are all made by our ancestors (we wouldn't exist without them); and if we had different ancestors we'd be a different person.  But,  as we are individual human beings, it would be false to say "a person is defined entirely by what ancestors she has".
