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Set theory is the mathematical theory of collections of objects, which can be numbers, characters, or even other sets. These objects are known as elements or members of a set. Set theory serves as a foundation for various fields within mathematics, providing a fundamental language and structure within which mathematical concepts can be formulated and related. An essential operation within set theory is the formation of a union of sets.
In set theory, the union of two sets A and B, denoted as \(A \bigcup B\), is a set that contains all of the elements that are in set A, set B, or in both. When we want to find the cardinal number, or the size, of this union, we seek to count all distinct elements without duplication. This concept is central when combining different sets, ensuring a proper understanding of how individual elements contribute to the total count. The ability to calculate the cardinal number of a union of sets is a vital skill in set theory, as it helps to quantify and compare different collections of objects.

Finite sets are those collections that have a bounded number of elements. Counting the number of elements in a finite set is straightforward compared to an infinite set which cannot be counted by conventional means due to their unbounded nature. One of the first steps in understanding the union of finite sets is to individually count the elements within each set, typically denoted as \(|A|\) for set A and \(|B|\) for set B.
Because we deal with finite sets, the concept of cardinality directly applies and becomes a measure of the set's size. In calculating the cardinal number for a union, the distinction between finite and infinite sets is crucial; the accompanying solutions and calculations are different for infinite sets. When exploring the intersection and union of finite sets, recognizing this limitation on the size of the sets is a key aspect of grasping the underlying principles of set theory.

The intersection of sets is another fundamental concept in set theory, denoted as \(A \bigcap B\). This describes a new set comprising all the elements that both set A and set B have in common. It plays a crucial role when determining the cardinal number of a union of finite sets.
Understanding the intersection is vital because, when counting the combined cardinal number of two sets, we may inadvertently count shared elements twice. To avoid this and find the accurate total number of distinct elements in the union, we must subtract the number of shared elements—the cardinal number of the intersection. This adjustment is mathematically expressed in the formula \(|A \bigcup B| = |A| + |B| - |A \bigcap B|\), which effectively ensures each element is counted only once. The intersection's cardinal number gives us the exact number of elements to subtract from the sum of the cardinalities of each set, thereby providing a true count for the union.