91Ó°ÊÓ

The term 'cardinality' may sound complicated, but it's really just a fancy word for the number of elements contained within a set. When we use notation like \( n(A) \), we are referring to the cardinality of set \( A \). For example, if set \( A \) has 15 distinct elements, we write \( n(A) = 15 \).

In problems involving set theory, figuring out the cardinality of a set is often a key step. Understanding cardinality is essential because it helps us quantify and compare different sets. For instance, knowing that \( n(B) = 20 \) tells us exactly how many elements are in set \( B \), which can be useful for solving complex problems or proving certain properties about the sets.

Now, let’s consider the 'intersection' of sets, which is denoted by \( A \cap B \). The intersection consists of all elements that are common to both sets. To visualize it, imagine two overlapping circles in a Venn diagram; the area they overlap represents the intersection.

For example, if we have two sets, Set \( A \) and Set \( B \), and they have five items in common, we denote this as \( n(A \cap B) = 5 \). The concept of intersection is fundamental in problems where we need to identify shared characteristics or elements among different sets. It allows us to see how groups are related to one another through their commonalities.

In contrast to the intersection, the 'union' of two sets, \( A \cup B \), is the set containing all the elements from both \( A \) and \( B \), with duplicates removed. It's like taking everything two sets can offer and putting it in one basket. To illustrate the union, you can again think of a Venn diagram, but this time, it's the total area covered by both circles, without any overlap.

An important point to remember is that while combining the sets, if there are common elements, we count them only once. For example, with \( n(A \cup B) = 30 \), we understand that there are 30 unique elements when both sets are considered together. Union is crucial for evaluating the full scope of different categories or topics when they are combined.

Grasping set theory requires familiarity with its fundamental formulas. These equations help us solve for unknowns within set problems. A vital formula to remember is for the union of two sets, expressed as \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \). This formula signifies that the total number of elements in the union of sets \( A \) and \( B \) is the sum of the cardinalities of each set, minus the number of elements they have in common.

Using this formula, as in our exercise, helps us unravel the cardinality of one set when we know the union and the intersection in relation to another set. It's a supremely useful tool that allows us to understand the full landscape of the sets without painstakingly listing all their elements. Knowledge of this and other formulas lays the groundwork for more advanced studies in mathematics.