Problem 10 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(A\) and \(B\) be subsets of a universal set \(U\) and suppose \(n(U)=200, n(A)=100, n(B)=80\), and \(n(A \cap B)=40\). Compute: a. \(n\left(A^{c} \cap B\right)\) b. \(n\left(B^{c}\right)\). c. \(n\left(A^{c} \cap B^{c}\right)\)

Step-by-Step Solution

Answer

a. \(n\left(A^{c} \cap B\right) = 40\) b. \(n\left(B^{c}\right) = 120\) c. \(n\left(A^{c} \cap B^{c}\right) = 80\)

Step 1: Understand the given information

We are given that \(A\) and \(B\) are subsets of the universal set \(U\). The number of elements in each set is as follows: - \(n(U) = 200\) - \(n(A) = 100\) - \(n(B) = 80\) - \(n(A \cap B) = 40\)

Step 2: Calculate n(\(A^{c} \cap B\))

We know that \(n(A \cap B) + n(A^{c} \cap B) = n(B)\). Therefore, we can find the number of elements in \(A^{c} \cap B\) using these given values. \[n(A^{c} \cap B) = n(B) - n(A \cap B) = 80 - 40 = 40\] So, \(n\left(A^{c} \cap B\right) = 40\).

Step 3: Calculate n(\(B^{c}\))

We know that \(n(U) = n(B \cup B^{c})\) and \(n(B \cup B^{c}) = n(B) + n(B^{c})\). Therefore, we can find the number of elements in \(B^{c}\) using these given values. \[n(B^{c}) = n(U) - n(B) = 200 - 80 = 120\] So, \(n\left(B^{c}\right) = 120\).

Step 4: Calculate n(\(A^{c} \cap B^{c}\))

We know that \(n(U) = n(A \cup A^{c})\) and \(n(A \cup A^{c}) = n(A) + n(A^{c})\). Therefore, we can find the number of elements in \(A^{c}\) using these given values. \[n(A^{c}) = n(U) - n(A) = 200 - 100 = 100\] Now, we can apply the principle of inclusion-exclusion to calculate the number of elements in \(A^{c} \cap B^{c}\): \[n(A^{c} \cap B^{c}) = n(A^{c}) + n(B^{c}) - n\left((A^{c} \cap B^{c})^{c}\right)\] Note that \((A^{c} \cap B^{c})^{c} = A \cup B\), so we have: \[n(A^{c} \cap B^{c}) = n(A^{c}) + n(B^{c}) - n(A \cup B)\] And by using the formula for the union of two sets, we get, \[n(A^{c} \cap B^{c}) = n(A^{c}) + n(B^{c}) - (n(A) + n(B) - n(A \cap B))\] Now, substitute the given values: \[n(A^{c} \cap B^{c}) = 100 + 120 - (100 + 80 - 40) = 220 - 140 = 80\] So, \(n\left(A^{c} \cap B^{c}\right) = 80\).

Key Concepts

Universal Set

In set theory, the **universal set** is the superset that contains all elements of any set being discussed. It acts as a larger whole from which subsets are derived. Any subset, such as set \(A\) or set \(B\), includes items that are also part of this universal set.

• The **notation** for the universal set is usually \(U\).
• In our example, \(U\) is said to have 200 elements, represented by \(n(U)=200\).

Understanding the universal set is crucial for determining complements and other subset relationships since it defines the bounds within which these sets operate.

Subsets

A **subset** is a set whose elements are all contained within another set, called the **universal set**.

• For example, if \(U\) is the universal set and \(A\) is a subset, all elements of \(A\) are in \(U\).
• In the problem, both \(A\) and \(B\) are subsets of \(U\), with \(n(A)=100\) and \(n(B)=80\).

Subsets help in organizing data and forming intersections or complements, which are used to solve various mathematical problems.

Intersection of Sets

The **intersection of sets** involves elements that are common to multiple sets.

• For two sets \(A\) and \(B\), their intersection is denoted as \(A \cap B\).
• This intersection tells you which elements exist in both sets.

In our example, the intersection \(n(A \cap B)=40\) is how many elements are common between sets \(A\) and \(B\). Intersections are crucial for understanding overlap among sets and are foundational to set operations in mathematics.

Complement of a Set

The **complement of a set** includes all the elements in the universal set \(U\) that are not part of a particular subset.

• If \(A\) is a subset of \(U\), the complement of \(A\), denoted \(A^c\), includes all elements in \(U\) minus those in \(A\).
• Similar logic applies to \(B^c\), the complement of set \(B\).

In given problems, it is often necessary to compute complements, such as \(n(B^c) = 120\) indicating the elements in \(U\) but not in \(B\). Calculating set complements aids in resolving how subsets differ from the universal set and each other, enhancing how overlaps and distinctions are managed.