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In set theory, the concept of a cardinal number helps us understand the size of a set by counting the number of elements it contains. This is a fundamental idea because it allows us to compare different sets by their sizes, regardless of the nature of their elements.

Consider set A from the exercise. It contains a total of 12 numbers and 18 letters. To find its cardinal number, we simply add these two values. Thus, the cardinal number \(|A|\) is 12 + 18 = 30. Similarly, for set B, which has 14 numbers and 10 letters, the cardinal number \(|B|\) is 14 + 10 = 24.

Cardinal numbers are critical when we need to perform operations involving set unions and intersections, as they give us a clear numerical picture of the sets in question.

The union of two sets A and B, written as \(A \cup B\), is a new set that includes all the elements that are in either set A, set B, or both. To find the number of elements in this union, we utilize the formula for the cardinal number of the union of two sets:

• \(|A \cup B| = |A| + |B| - |A \cap B|\)

This operation is essential in mathematical problem solving because it helps us quantify how many unique elements exist between the two sets.

In the exercise, we calculated \(|A \cup B|\) using the formula and identified that there are 47 unique elements in the union of sets A and B. This result was achieved by combining the total elements of each set and subtracting the common elements counted twice.

The intersection of two sets A and B, denoted as \(A \cap B\), contains all the elements common to both sets. It's a useful way to discern commonalities between two data sets, which can be especially useful in scenarios involving overlapping data.

In the given problem, we identified that the intersection \(A \cap B\) contains one number and six letters. This means \(|A \cap B|\) is 1 + 6 = 7.

The ability to accurately find the intersection of sets is vital in mathematical problem solving, as it allows for the precise calculation of the union of sets, helping to efficiently organize and classify data.

Mathematical problem solving often involves applying known formulas and theories to new situations to derive solutions. In set theory, understanding and applying the formulas for cardinal number, union, and intersection of sets are crucial steps.

For the exercise at hand, we needed to determine the number of elements in the union of two sets. The problem was successfully solved by carefully counting the elements in each set and correctly applying the formula for the union of sets.

This approach reinforces the importance of systematic and logical thinking in mathematics. By following a step-by-step process, we can tackle complex problems, ensuring the correct application of mathematical concepts and deriving accurate solutions.