91Ó°ÊÓ

Set theory is a fundamental part of mathematics that deals with the collection of objects, referred to as elements or members of the set. In the context of the exercise, we can think of each group of freshmen who attended the movie showings as different sets.

For instance, let's define set A as the group of students who attended the first showing and set B as those who attended the second showing. The entire freshman class is the universal set, often denoted by U, while the subset of students who didn't attend either showing can be represented as the complement of the union of sets A and B, written as \( U - (A \cup B) \).

In terms of set operations, finding out how many students attended both showings involves looking for the intersection of A and B, denoted by \( A \cap B \). Comprehending these basic set operations and representations is crucial to solving problems involving groups and their interactions.

The inclusion-exclusion principle is a vital concept in combinatorial mathematics, particularly when it comes to counting the elements in the union of sets. This principle allows us to calculate the number of elements in the union of multiple sets without overcounting the elements that appear in more than one set.

In the exercise, we apply the inclusion-exclusion principle to avoid overcounting the students who attended both movie showings. The principle tells us that to find the total number of unique attendees (the union of set A and set B), we must add the number of attendees for the first showing (A) to the number of attendees for the second showing (B) and then subtract the number of attendees that were counted in both showings (\( A \cap B \) or the intersection of A and B).

The principle's general formula for two sets is: \[ |A \cup B| = |A| + |B| - |A \cap B| \] where \( |X| \) denotes the number of elements in set X. Applying this principle correctly is essential for solving the given problem and many others in combinatorial mathematics.

Mathematical problem-solving is about understanding the problem, devising a plan, carrying out the plan, and reviewing the solution. In our James Bond movie example, understanding the problem involves recognizing the sets involved and their relationships.

The problem was approached by first identifying the known variables (total freshmen, attendees at each showing, and non-attendees). The next step was to use these variables in a plan that incorporated set theory and the inclusion-exclusion principle to calculate the number of freshmen who attended both showings.

Finally, reviewing the solution involves verifying that the numbers make sense within the context of the problem. For instance, the number of students who attended both showings (55) added to those who didn't attend at all (450) should not exceed the total number of freshmen (600). Enabling students to understand and apply this systematic process will significantly enhance their problem-solving skills in mathematics and beyond.