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In set theory, the concept of the union of sets is crucial. Imagine you have different groups, like baskets of fruits. The union of these groups would be like gathering all the fruits from each basket into one big basket. In mathematical terms, the union creates a new set containing all the unique elements from the original sets.

You use the union operation to combine multiple sets. The symbol for union is \( \cup \). When you are asked to find the union of sets, you essentially list every element from all the specified sets, making sure not to list any element more than once.

In the exercise given, to find the union of \(E\), \(F\), and \(G\), you would take each of the numbers from these sets, and place them into a new set without repeating any number:

• \(E = \{2, 4, 6\}\)
• \(F = \{1, 3, 5\}\)
• \(G = \{5, 6\}\)

The union of these sets is \( E \cup F \cup G = \{1, 2, 3, 4, 5, 6\} \). This represents a set containing each different element found in the three sets combined.

Disjoint sets are sets that have no elements in common. When you look at two or more sets, if there are no shared elements across them, they are called disjoint.

To understand this with an example, think of two circles that don't overlap. Each set is like a circle, and because they don't touch or intersect, there's no "shared space" or common elements.

In the original exercise, none of the sets \(E\), \(F\), and \(G\) are completely disjoint pairs because they do share some elements. For example, \(F\) and \(G\) both include the number 5, so they share this element and therefore are not disjoint.

It's important to recognize disjoint sets, as understanding them can help simplify many mathematical problems involving set operations. When sets are disjoint, the union is simply a combination of all elements since none will overlap, making calculation straightforward.

When dealing with sets, it's important to understand how to handle element repetition. In a set, every element is unique, meaning you should never list the same element more than once.

This property makes set operations quite straightforward. You only need to include each number once, regardless of how many times it appears in different sets.

Take the exercise step where we found the union \(E \cup F \cup G\). Although there are repeated elements like 5 in both \(F\) and \(G\), and 6 in \(E\) and \(G\), they are only listed once in the union result:

\[E \cup F \cup G = \{1, 2, 3, 4, 5, 6\}\]

This rule simplifies working with sets by ensuring that each number is distinctly counted, eliminating the need for duplication in the output set.