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The concept of the union of sets is akin to bringing together different groups and finding what they all share collectively, but without repeating any members. When we perform a union, designated by the symbol \(\bigcup\), we're essentially combining all the unique elements from each set involved.

For example, in the exercise where we have sets \(E = \{2, 4, 6\}\), \(F = \{1, 3, 5\}\), and \(G = \{5, 6\}\), their union \(E \bigcup F \bigcup G\) includes every distinct number from these sets, which results in \(\{1, 2, 3, 4, 5, 6\}\). As you can notice, even though the number 5 appears in both sets \(F\) and \(G\), it is only counted once in the union set. This elimination of duplicates is a fundamental rule in forming the union of sets.

Imagine you have a complete set, let's call it a 'universal set', which contains all the possible elements under consideration. The complement of a set, symbolized by a superscript \(c\), is a way of finding out what's not included in a particular subset of this universal set. It's like looking at a puzzle and figuring out which pieces are missing to complete the picture.

In our exercise, the universal set is \(S = \{1, 2, 3, 4, 5, 6\}\), and after finding the union \((E \bigcup F \bigcup G)\), we discovered it's exactly the same as the universal set \(S\). Therefore, when we look for the complement \((E \bigcup F \bigcup G)^c\), we're searching for elements in \(S\) not found in the union. In this case, since the union covers all of \(S\), the complement is an empty set, represented by \(\emptyset\). This signifies that there are no pieces missing; our puzzle is complete.

Finite mathematics covers a host of topics, but it's primarily known for dealing with objects that you can count using whole numbers. It doesn't concern itself with the notion of infinity; everything has a limit or an end. Within the realm of set theory in finite mathematics, understanding how sets interact through union and complement operations is key.

Set operations like these provide a foundation for various aspects of finite mathematics, including probability, statistics, and even discrete math. The exercise demonstrates how finite set operations work in practice, as we discovered the outcome of a union followed by a complement within a finite set \(S\). With these building blocks, finite mathematics can tackle more complex problems, all while remaining in the realm of the countable and the tangible.