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The cardinality of a set refers to the number of elements contained within that set. It is a measure of the \'size\' of the set. Cardinality is denoted by the letter \'n\', followed by the set in parentheses, such as in the expression \(n(A)\) for set A. To find the cardinality, you simply count the distinct elements present.

For example, in the provided exercise, we calculated the cardinality of set A, which contains the vowels: \( A = \{a, e, i, o, u\} \), and determined that \(n(A) = 5\). Similarly, we determined the cardinality of set B, with the consonants: \( B = \{g, h, k, l, m\} \), as \(n(B) = 5\). Understanding the cardinality of a set is essential when comparing sets or performing operations such as unions or intersections.

The union of two or more sets consists of all the distinct elements that appear in any of the sets. Mathematically, we denote the union of sets A and B as \(A \cup B\). If an element is part of either set A or set B, or both, it will be included in the union of these sets.

In the case of the exercise, since sets A and B are disjoint, we know they share no common elements, which simplifies the process of forming their union. Therefore, \(A \cup B\) will encompass all the elements from both A and B without any overlap. As demonstrated in the solution, \(A \cup B = \{a, e, i, g, h, k, l, m, o, u\}\), which includes all elements from both sets combining into a larger set with a total cardinality of ten.

Disjoint sets are sets that do not share any elements, meaning they have no overlap. When two sets are disjoint, their intersection, the set of elements they have in common, is empty. This concept is essential when considering union operations because the union of disjoint sets will simply be the combination of all elements from both sets without duplications.

In our exercise, the sets A (vowels) and B (consonants) are perfect examples of disjoint sets, as there is no letter that is both a vowel and a consonant. Acknowledging that the sets are disjoint confirms that their union will have a cardinality equal to the sum of the individual cardinalities; hence, \(n(A \cup B) = n(A) + n(B)\) as illustrated previously.

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. In applied mathematics, the set theory concept is often used to solve problems involving data classification, statistics, probability, and more. It aids in understanding the relationship between different data groups or sets, as seen in the textbook exercise with disjoint sets. By calculating the cardinality of sets and their unions, students embody the role of applied mathematicians, handling real-world problems that can be abstracted and processed through mathematical concepts.

Understanding how set theory pertains to applied mathematics gives students a practical framework to apply these skills beyond the classroom, fostering analytical and problem-solving abilities that are invaluable in various professions and in research.