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Set theory is the mathematical science of sets, where a 'set' is defined as a collection of distinct objects, considered as an object in its own right. These objects are called 'elements' or 'members' of the set. Sets are often denoted using curly braces, like so: \( \{1, 2, 3\} \). Elements within a set are unordered and unique, which means the set \( \{1, 2, 3\} \) is the same as \( \{3, 1, 2\} \), and you won't find duplicates within one set.

Understanding set theory is vital because it serves as the foundation of various fields within mathematics, including algebra, probability, and many others. This theory provides a standard way to define and study collections of objects, whether they are numbers, geometric shapes, or even more abstract concepts. When we discuss the union of sets, we're implementing one of the primary operations set theory offers.

Venn diagrams are a tool used to visually represent sets and the relationships between them. They are especially useful when it comes to depicting operations like unions, intersections, and complements. A Venn diagram usually consists of overlapping circles, where each circle represents a set. The overlapping areas indicate where sets have elements in common.

In the case of a union, which encompasses all the elements from each set involved, the Venn diagram would show both the non-overlapping and overlapping sections of the circles filled in. This visual representation helps students grasp the concept of a union, which boils down to combining different sets and including all distinct elements from each, as done in the exercise with sets \( \{1,2,3,4\} \) and \( \{2,4,5\} \). A Venn diagram for this union would show circles intersecting at elements 2 and 4, with 1, 3, and 5 unique to their respective sets.

Elementary set operations are the basic ways in which we can combine, compare, and analyze sets. The three primary operations are union, intersection, and set difference. The union of two sets, usually denoted as \( A \cup B \), is a set that contains all the elements of set A and set B without any duplicates.

For instance, when combining the sets \( \{1,2,3,4\} \) and \( \{2,4,5\} \), the union operation instructs us to consider every element from both sets while ensuring each element is unique. Therefore, the result of this union is \( \{1, 2, 3, 4, 5\} \).

Understanding how to merge sets and remove duplicates is essential, as it's not just about creating a list of elements but about forming a new set that adheres to the definitions provided by set theory.