quick find algorithm - union operation - is it same as union in set theory?

Question

I am unable to correlate union operation in quick find algorithm with general meaning of A U B in set theory.

Book(Algorithms in C++ Robert Sedgewick) tells union operation is "scanning throgh the whole array for each input pair.(line 9 and 10 in code).

Basically we are copying value at node q to all other nodes having same value as node p. Why do we name this operation as UNION?

Code is directly copied from the book.

#include <iostream>
const int N = 10000;
int main() {
int i, p, q, id[N];
for( i = 0; i < N; i++ ) id[i] = i;
while( cin >> p >> q ) {
    int t = id[p];
    if ( t = id[q] ) continue;              //quick find operation
    for ( i = 0; i < N; i++ )               //---> union why?
        if ( id[i] == t) id[i] = id[q];
    cout << " " << p << " " << q << endl;
}
}

Answer 1

The union step in quick find means merging the components with the same id. In a general sense it's kinda like the union of two sets. You can consider two sets, onw with id1 as id of all its components and another as id2. For a great explanation look at this presentation in the quick find section:

http://www.cs.princeton.edu/~rs/AlgsDS07/01UnionFind.pdf

Answer 2

Look at the set of operations supported. If there is no way to ask "list all the elements", but just insert, find & union, then there is no way to tell, using these operations, whether or not there is duplication of elements. It makes the supported operations more efficient, and still BEHAVES (as far as the user can tell) like a set.