why I cannot change the value in loop of `set<set<int>>` - c++

Question

Here is the minimum function of my question:
void solve(void)
{
set<set<int> > test;
for(set<set<int> >::iterator it = test.begin(); it != test.end(); ++it) {
// (*it)'s type is set<int> right? but why I cannot insert a int into this?
it -> insert(1);
}
}
The compiler tell me that no matching function for call to 'std::set<int>::insert(int) const'.
It really confuse me that why it do not use std::set<int>::insert(int) without the const? How can I do this? Can I make it really?
*** first edited ***
I know that I can erase an element like it = test.erase(it) and I also know that I can use test.insert(<a set<int> value>), so is it possible to insert firstly and then delete the original element? but it looks troublesome right?
*** seconed edited ***
About the use-case: Before I know the Disjoint-set data structure, I want to slove its problem, so I need to deal with the set<int> type data in set<set<int>> type data. It upset me that I find I cannot change the inner set in the loop so I cannot union those inner sets - (or maybe I can, I try to put those elements of inner set's value into a temp set and then I remove the original sets, then I insert the temp set into set<set>. BUT it does look ugly). So I know to make it but I do not know why the iterators are const, and I also want t know a better way to deal with sets in set.
*** third edited ***
Thank you #anurag-dhadse for pointing the grammar error.
Thanks.

Answer 1

Your it->insert(1) attempts to change a set<int> inside the outer set<set<int>>, and in so doing might change the position in which *it should be stored in the outer set, which would breach the class invariants by not keeping the elements sorted. To avoid that, the outer set only gives it const access to the set<int> elements
If you want to modify a set<int> element, you need to extract it from the outer set, modify it, then insert it back wherever it should now go.

Answer 2

You realize that if you insert many sets inside an "outer" set, those inner sets will be placed in lexicographical order:
#include <set>
#include <vector>
#include <iostream>
using namespace std;
int main () {
set A1 = {2, 7};
set A2 = {1, 8, 3, 4};
set A3 = {1, 4, 3};
set<set<int>> set_of_sets;
set_of_sets.insert(A1);
set_of_sets.insert(A2);
set_of_sets.insert(A3);
for ( auto& inner_set : set_of_sets ) {
std::cout << "[ " ;
for ( auto &element : inner_set ) {
std::cout << element << " ";
}
std::cout << "]\n";
}
}
Will produce the following response:
[ 1 3 4 ]
[ 1 3 4 8 ]
[ 2 7 ]
They changed the order, not only the integers inside each set, but A1 was swapped by A3 (following lexicographical order )
https://godbolt.org/z/nacxrn
If you insert 1 to each set, the [1 3 4] will not change, the [1 3 4 8] will also not change, but [2 7] will become [1 2 7] wich means that A1 should be swappped by A3 again since [1 2 7] < [1 3 4], therefore C++ does not allow you to change the position of the inner sets inside the outer set during iteration ( by making iterators const ).
So, it´s not clear why you are using a set.
it´s easy ( for many applications ) just use a vector of sets so you can change them freely:
vector<set<int> > test { A1, A2, A3 } ;
for(auto& s : test ) {
s.insert(1); /// inserting 1 in all sets
}
https://godbolt.org/z/WMvse9