Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(A, B\), and \(C\) be subsets of a universal set \(U\) and suppose \(n(U)=100, n(A)=28, n(B)=30\) \(n(C)=34, n(A \cap B)=8, n(A \cap C)=10, n(B \cap C)=15\) and \(n(A \cap B \cap C)=5\). Compute: a. \(n(A \cup B \cup C)\) b. \(n\left(A^{c} \cap B \cap C\right)\)

Step-by-Step Solution

Answer

a. The cardinality of \(A \cup B \cup C\) is 60. b. The cardinality of \(A^c \cap B \cap C\) is 10.

Step 1: Apply the principle of Inclusion-Exclusion for three sets

We will use the principle of Inclusion-Exclusion to find the cardinality of \(A \cup B \cup C\). The formula is as follows: \[n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C)\] We are given the cardinalities of each individual subset and the cardinalities of every possible intersection, so we can simply substitute the given values into the formula:

Step 2: Substitute values and compute \(n(A \cup B \cup C)\)

\[\begin{aligned} n(A \cup B \cup C) &= n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \\ &= 28 + 30 + 34 - 8 - 10 - 15 + 5 \end{aligned}\] Now, compute the value: \[n(A \cup B \cup C) = 60\] For part (a), the cardinality of \(A \cup B \cup C\) is 60.

Step 3: Compute \(n\left(A^{c} \cap B \cap C\right)\)

In order to find \(n\left(A^{c} \cap B \cap C\right)\), we can first find the cardinality of \(B \cap C\), which is given as 15. Then, we need to subtract the cardinality of \(A \cap B \cap C\), which is 5, since we are only interested in elements that belong to \(B \cap C\) but not \(A\). Thus, the cardinality of \(A^c \cap B \cap C\) is: \[n\left(A^{c} \cap B \cap C\right) = n(B \cap C) - n(A \cap B \cap C) = 15 - 5\]

Step 4: Compute the final result for part (b)

\[n\left(A^{c} \cap B \cap C\right) = 10\] For part (b), the cardinality of \(A^c \cap B \cap C\) is 10.

Key Concepts

Set Theory

Set theory is a branch of mathematical logic that studies sets, which are collections of objects. These objects can be anything: numbers, letters, people, etc. In set theory, these collections are usually defined as distinct and separate entities. Consider a set as a container that holds these objects, often known as elements or members.
Set theory underpins most of modern mathematics, providing a solid foundation for defining numbers, geometry, and other concepts. If you think about sets in terms of Venn diagrams, where circles represent different sets, it becomes easier to visualize how sets can overlap or remain separate.
Important concepts in set theory include:

• Subsets: If every element of a set A is also in set B, then A is a subset of B.
• Union: The union of two sets includes any element that is in either one of the sets or in both.
• Intersection: The intersection of two sets includes only the elements that are present in both sets.
• Complement: Elements that are not in the set but belong to the universal set.

Understanding these basic principles of set theory is essential for grasping more complex topics like the Inclusion-Exclusion Principle.

Universal Set

The universal set, denoted by U, is the set that contains all the objects under consideration, or all possible elements for a particular discussion. For different contexts, the universal set can change depending on what elements are being considered.
In practical terms, the universal set is like the largest container in which the other subsets exist. Every set involved in a discussion or calculation is a subset of the universal set. For example, if we're discussing animals in a zoo, our universal set includes all the animals in that zoo.
The universal set is essential when we talk about complements or missing elements in a set. In set operations, it provides a complete frame of reference:

• The complement of a set A, written as A^c, includes all the elements in the universal set that are not in A.
• Notation is crucial here; understanding the role of the universal set helps clarify how different set operations interact.

The concept of a universal set provides a boundary within which mathematical operations on sets are considered.

Cardinality

Cardinality refers to the number of elements in a set. It's a measure of the "size" of the set, no matter what the elements are. For instance, if set A contains the numbers 1, 2, and 3, then the cardinality of set A, denoted by n(A), is 3.
Cardinality helps determine the possible configurations of set unions, intersections, and other operations. In problems involving the Inclusion-Exclusion Principle, knowing the cardinalities of various individual sets and their intersections helps us compute the cardinality of their union effectively.
Here are some important points regarding cardinality:

• If a set is empty, its cardinality is 0.
• The cardinality of a finite set is a non-negative integer.
• In more advanced mathematics, infinite sets have a cardinality that can be compared using the concept of 'infinite size'.

Understanding cardinality is crucial in set theory to allow for the effective application of strategies like the Inclusion-Exclusion Principle.

Intersection of Sets

The intersection of sets focuses on finding common elements between two or more sets. If we have two sets, A and B, then their intersection, denoted as A ∩ B, includes all elements that are common to both A and B.
In Venn diagrams, this is typically represented by the overlapping region of two circles. The concept of intersection helps solve problems where we need to find shared characteristics or elements, and it's widely used in probability, statistics, and data analysis.
Key points about intersections include:

• The intersection of two sets is itself a set.
• If two sets have no elements in common, their intersection is the empty set, ∅.
• For three or more sets, the intersection can be extended. For example, the intersection of sets A, B, and C is the set of elements present in all three sets A, B, and C.

The intersection of sets is a foundational concept for solving problems involving multiple criteria or parameters, where commonality is a focus.