Problem 14 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

If \(n(A)=16, n(B)=16, n(C)=14, n(A \cap B)=6\), \(n(A \cap C)=5, n(B \cap C)=6\), and \(n(A \cup B \cup C)=31\), find \(n(A \cap B \cap C) .\)

Step-by-Step Solution

Answer

The value of \(n(A \cap B \cap C)\) is 8.

Step 1: Write down the given information and the PIE formula

We are given that \(n(A)=16, n(B)=16, n(C)=14, n(A \cap B)=6\), \(n(A \cap C)=5, n(B \cap C)=6\), and \(n(A \cup B \cup C)=31\). We want to find \(n(A \cap B \cap C)\). We will use the Principle of Inclusion and Exclusion (PIE) formula: $$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)$$

Step 2: Plug in the given values into the PIE formula

Now let's plug in the given values into the formula: $$31 = 16 + 16 + 14 - 6 - 5 - 6 + n(A \cap B \cap C)$$

Step 3: Simplify the equation and solve for \(n(A \cap B \cap C)\)

Let's simplify the equation and solve for \(n(A \cap B \cap C)\): $$31 = 40 - 17 + n(A \cap B \cap C)$$ $$n(A \cap B \cap C) = 31 - 23 = 8$$ The value of the intersection of all the given sets, \(n(A \cap B \cap C)\), is 8.

Key Concepts

Set Theory

Set theory is a fundamental area of mathematics that deals with collections of objects, known as sets. A set can be anything: numbers, letters, shapes, or even other sets. The language of set theory is used to describe how these sets relate to each other.

For example, the sets in our exercise refer to collections of elements, where 'A', 'B', and 'C' can be thought of as different groups with varying number of members. In mathematical terms, the number of elements in a set is referred to as the cardinality of the set, and is denoted as 'n(A)' for set 'A'. Understanding set theory is crucial when trying to figure out relationships like unions and intersections that are part of this exercise.

Sets can be related in different ways and set theory gives us the tools—such as union, intersection, and set difference—to analyze these relationships. By grasping the concepts of set theory, we can better understand complex relations and solve problems that involve multiple sets, just like in the textbook exercise we are examining.

Intersection of Sets

The intersection of two or more sets is a new set containing all the elements that are common to all the original sets. In notation, we use the symbol \( \cap \) to denote the intersection. So the intersection of sets 'A' and 'B' is written as \( A \cap B \).

In our exercise, for instance, \( n(A \cap B) \) equals 6, which means there are 6 elements that both set 'A' and set 'B' have in common. Understanding this concept is vital as it allows us to identify shared elements between groups and often plays a key role when we want to determine the total number of distinct elements in a collection of sets.

Union of Sets

While intersections deal with commonalities, the union of sets is all about inclusivity. The union of two or more sets includes every element that is in either set, without duplication. The union is denoted by the symbol \( \cup \).

For example, the union of sets 'A', 'B', and 'C' is symbolized as \( A \cup B \cup C \). In our exercise, \( n(A \cup B \cup C) = 31 \) tells us there are 31 unique elements collectively in sets 'A', 'B', and 'C'. To accurately calculate the cardinality of a union of sets, we need to apply the Principle of Inclusion and Exclusion, ensuring that we count each element exactly once even if it appears in more than one set.

Cardinality of Sets

The cardinality of a set is the term used to describe the number of elements it contains. It is essentially a measure of a set's size. For any set 'X', its cardinality is denoted by \( n(X) \).

The exercise provides the cardinality for sets 'A', 'B', and 'C', as well as for their intersections. When we use the Principle of Inclusion and Exclusion, we effectively adjust for overcounting or undercounting caused by multiple counting of elements that are present in more than one set. By calculating the cardinality of various unions and intersections, we can piece together the cardinality of a more complex set, like \( n(A \cap B \cap C) \), which represents elements common to 'A', 'B', and 'C'.

Understanding cardinality is crucial for answering many real-world problems that pertain to quantity and organization, making it one of the pivotal aspects of set theory.