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Set theory is the mathematical study of collections of objects, known as sets. These objects are referred to as 'elements' or 'members' of the set. In set theory, a set is defined by its elements rather than the order or the number of times an element occurs.

For example, the set containing the numbers 1, 2, and 3 is written as \(\{1, 2, 3\}\). It's important to note that in set theory, repetition of elements is ignored. Thus, \(\{1, 1, 2, 3\}\) is considered the same as \(\{1, 2, 3\}\) because both represent a collection with the elements 1, 2, and 3. In problems involving set theory, various operations can be performed, including unions, intersections, and set differences.

Understanding the fundamental building blocks of set theory begins with the elements of a set. Each object contained in a set is called an element. Elements can be numbers, symbols, or even other sets. They must be well-defined and distinct within the context of the set. For instance, in the set \(\{a, b, c\}\), 'a', 'b', and 'c' are each elements of the set.

One thing to remember is that distinctness is paramount; if an element appears more than once in the listing of a set, it is still considered as one instance. For example, in the example from the exercise \(\{1,3,7,8\}\), '1', '3', '7', and '8' are the distinct elements of the set.

Combining sets is an essential operation in set theory, and one fundamental way to do this is through the union of two sets. The union of two sets \(A\) and \(B\), denoted as \(A \cup B\), is a new set containing all the elements that are in set \(A\), in set \(B\), or in both.

How to Form the Union of Two Sets

Following the steps from our exercise, to find the union you first identify all the elements present across both sets. For example, if you have two sets \(A = \{1,3,7,8\}\) and \(B = \{2,3,8\}\), you list out all elements without regard to which set they come from: \(1, 2, 3, 7, 8\).

After listing, you remove any duplicate elements since a set is defined as a collection of unique elements. The resulting set from our example would be \(\{1, 2, 3, 7, 8\}\), which represents the union of the two sets \(A\) and \(B\). The concept of union is vital when combining multiple sets to understand the complete range of distinct elements present in said sets.