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The union of sets is a fundamental concept in set theory. It involves combining all the elements of two or more sets into a single set. The union denoted by the symbol \( \cup \) includes every element that is in either of the original sets, without repeating any elements. For example, if you have sets \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), their union \( A \cup B = \{1, 2, 3, 4, 5\} \). Notice that the element "3" appears only once in the union, even though it is present in both sets. This is because each element is counted only once in a union.In the exercise, the union of sets \( A \) and \( B \) was determined using a formula for the cardinal number of the union. This formula accounts for any overlap between sets to ensure elements aren't counted multiple times.

Set theory serves as the foundation for modern mathematics. It provides the language and tools to discuss collections of objects, known as sets. Sets are versatile and can represent various things like numbers, shapes, or even other sets. In set theory, we use symbols and operations to describe relationships between sets, such as union, intersection, and difference. A crucial aspect of sets is the notion of cardinality, which represents the number of elements in a set. The cardinality of a set \( A \) is denoted as \( |A| \). When learning set theory, it's beneficial to become familiar with Venn diagrams. These diagrams visually represent the relationships between sets, illustrating operations like unions or intersections. Understanding how elements are shared between sets or are unique to each set is key to mastering concepts such as unions, intersections, or complements.

The intersection of sets focuses on common elements shared between two or more sets. The result of an intersection is a new set, which contains only the elements that are present in each of the original sets. The symbol for intersection is \( \cap \). For instance, if \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), their intersection \( A \cap B = \{3\} \), because "3" is the only element found in both sets. Intersections help us identify what overlap exists in terms of shared elements.In the given exercise, recognizing the intersection between sets \( A \) and \( B \), with 5 common elements, was crucial to solving the problem correctly. By using these shared elements, we can apply the formula for the union's cardinality to avoid counting elements more than once in the union.