Question

At the beginning of the semester, an instructor of a music appreciation class wants to find out how many of the 250 students had heard recordings of the music of Mozart. Becthoven, Haydn, or Bach. The survey showed the following: How many students had listened to none of the composers? $$\begin{array}{||l|c|} \hline \text { Composer Listened to by Students } & \text { No. of Students } \\\ \hline \text { Mozart } & 125 \\ \hline \text { Beethoven } & 78 \\ \hline \text { Haydn } & 95 \\ \hline \text { Bach } & 62 \\ \hline \text { Mozart and Beethoven } & 65 \\ \hline \text { Mozart and Haydn } & 50 \\ \hline \text { Mozart and Bach } & 48 \\ \hline \text { Beethoven and Haydn } & 49 \\ \hline \text { Beethoven and Bach } & 39 \\ \hline \text { Haydn and Bach } & 37 \\ \hline \text { Mozart, Beethoven, and Haydn } & 22 \\ \hline \text { Mozart, Beethoven, and Bach } & 19 \\ \hline \text { Mozart, Haydn, and Bach } & 18 \\ \hline \text { Beethoven, Haydn, and Bach } & 13 \\ \hline \text { Mozart, Beethoven, Haydn, and Bach } & 9 \\ \hline \end{array}$$

Short answer

The Inclusion-Exclusion Principle is a method used to calculate the number of elements in the union of several sets by including and excluding intersections of these sets. For four sets, the formula is: \[ |A \cup B \cup C \cup D| = |A| + |B| + |C| + |D| - |A \cap B| - |A \cap C| - |A \cap D| - |B \cap C| - |B \cap D| - |C \cap D| \]\[ + |A \cap B \cap C| + |A \cap B \cap D| + |A \cap C \cap D| + |B \cap C \cap D| \]\[- |A \cap B \cap C \cap D| \]

Step-by-step solution

Understand the Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a method used to calculate the number of elements in the union of several sets by including and excluding intersections of these sets. For four sets, the formula is: \[ |A \cup B \cup C \cup D| = |A| + |B| + |C| + |D| - |A \cap B| - |A \cap C| - |A \cap D| - |B \cap C| - |B \cap D| - |C \cap D| \]\[ + |A \cap B \cap C| + |A \cap B \cap D| + |A \cap C \cap D| + |B \cap C \cap D| \]\[- |A \cap B \cap C \cap D| \]

Key concepts

Set Theory

Set Theory is a foundational discipline in mathematics that deals with the collection of objects, considered as sets. In Set Theory, a set is defined as a well-defined collection of distinct objects, known as elements. For instance, a set could be written as \( A = \{1, 2, 3, 4\} \), where \(1, 2, 3,\) and \(4\) are the elements of the set \(A\).
Sets are generally depicted using symbols like \(A, B, C,\) and so on.In our specific case, we have a number of students each associated with different composers' music they have heard. These students can be grouped into sets based on the composer. The fundamental operations in Set Theory help us to determine how many students listened to the music of various combinations of these composers.
Through applying the Inclusion-Exclusion Principle, which is deeply rooted in set operations, we can explore how these sets interact with each other.

Union of Sets

The union of sets is a fundamental concept in Set Theory. The union, symbolized by \( \cup \), represents the set containing all distinct elements belonging to any of multiple sets. For example, if set \(A = \{1, 2\}\) and set \(B = \{2, 3\}\), then the union \(A \cup B = \{1, 2, 3\}\).
In the context of our problem, the union of sets would mean gathering all students who have heard music from any combination of the composers listed: Mozart, Beethoven, Haydn, and Bach.
The Inclusion-Exclusion Principle is essential here because when you simply add up all the individual students for Mozart, Beethoven, Haydn, and Bach, you overcount the students who fall into multiple categories unless you carefully apply corrections for intersections.

Intersections of Sets

The intersection of sets is another key operation in Set Theory, involving the elements common to all sets involved. This operation is symbolized by \( \cap \). For example, the intersection for sets \(A = \{1, 2\}\) and \(B = \{2, 3\}\) is \(A \cap B = \{2\}\) because '2' is the only element present in both sets.
In our problem, intersections help identify students who listened to music from multiple composers. These are crucial because they help us adjust our count when using the Inclusion-Exclusion Principle. By determining these intersections—such as those who have heard both Mozart and Beethoven—we recognize and adjust our total number to not count these students more than once.

Counting Problems

Counting problems often arise in mathematics, requiring strategies to accurately tally elements in combined groups or sets. With complex overlapping of groups, like in the student's composer listening habits here, a straightforward sum isn't enough.
That's where the Inclusion-Exclusion Principle comes in, allowing us to account for overcounts in overlapping members.
Here's how it works:

• First, add up the students who have listened to each composer individually.
• Subtract the counts of students who have listened to each pair of composers, since they have been added twice initially.
• Add back the counts for those who listened to three composers, correcting for over-subtractions.
• Subtract those who have heard all four composers, balancing their fourfold counting previously.

These adjustments capture the correct number of students who have listened to music from any of the composers and those who haven't. Understanding this structure and application is key to solving many counting problems, especially those involving complex group overlaps.