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The union of sets is a fundamental concept in set theory. It involves combining all the unique elements from two or more sets into a new set. This new set, known as the union, includes every element that appears in any of the original sets.

To denote the union, we use the symbol \( \cup \). For example, if we have sets \( A \) and \( B \), the union \( A \cup B \) includes all elements that are in \( A \), in \( B \), or in both.

In our example:

• Set \( A = \{-10, -9, -8\} \)
• Set \( B = \{8, 9, 10\} \)

Since there are no elements in common between \( A \) and \( B \), the union \( A \cup B = \{-10, -9, -8, 8, 9, 10\} \) is simply a combination of all the elements from both sets without any repeats.

Remember, the union set will always have unique elements, meaning each element from the original sets appears only once in the union.

In set theory, elements refer to the distinct objects or members that make up a set. It is important to understand that each element in a set is unique, which means it cannot be repeated within the same set.

For example, consider set \( A = \{-10, -9, -8\} \). The elements of Set \( A \) are \(-10\), \(-9\), and \(-8\). Each of these numbers is an individual element in the set.

Similarly, in set \( B = \{8, 9, 10\} \), the elements are \(8, 9,\) and \(10\). These numbers represent distinct elements in Set \( B \).

In any union operation, as seen with \( A \cup B \), all elements from both sets are combined, but each is only listed once, regardless of whether it appears in more than one set.

This principle is key to understanding how the union of sets works, ensuring each element is accounted for and contributes to the complete set without duplication.

Disjoint sets are a special type of sets in set theory where the sets have no elements in common. In other words, if you were to compare two disjoint sets, you would find that their intersection is empty, meaning there are zero shared elements.

For instance, consider the sets from our exercise:

• Set \( A = \{-10, -9, -8\} \)
• Set \( B = \{8, 9, 10\} \)

As we can see, there isn't a single number that appears in both sets.

Because of this, sets \( A \) and \( B \) are disjoint. The union of two disjoint sets is straightforward to determine. It simply involves combining all elements from both sets into one, encompassing all unique elements without repeats.

Understanding disjoint sets is crucial as it frequently appears in problems involving set operations and helps simplify problems where you need to clearly delineate where overlap does or does not occur.