Problem 43 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(U\) denote the set of all employees in a hospital. Let $$ \begin{array}{l} \boldsymbol{N}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a nurse }\\} \\ \boldsymbol{D}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a doctor\\} } \\ \boldsymbol{A}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is an administrator\\} } \\ \boldsymbol{M}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a male\\} } \\ \boldsymbol{F}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { is a female }\\} \end{array} $$ Describe each set in words. a. \(D^{c}\) b. \(N^{c}\)

Step-by-Step Solution

Answer

a. \(D^{c}\) includes all employees who are NOT doctors (nurses, administrators, and any other staff members, regardless of their gender). b. \(N^{c}\) includes all employees who are NOT nurses (doctors, administrators, and any other staff members, regardless of their gender).

Step 1: a. Finding \(D^{c}\) - The Complement of Set D

To find the complement of set D, denoted as \(D^{c}\), we need to look for all the elements in the universal set U that are not in set D.

Step 2: aa. Describing \(D^{c}\) - The Complement of Set D in Words

Since set D contains all the doctors in the hospital, the complement (\(D^{c}\)) would include all employees who are NOT doctors. This will include nurses, administrators, and any other staff members who are not doctors, regardless of their gender.

Step 3: b. Finding \(N^{c}\) - The Complement of Set N

To find the complement of set N, denoted as \(N^{c}\), we need to look for all the elements in the universal set U that are not in set N.

Step 4: bb. Describing \(N^{c}\) - The Complement of Set N in Words

Since set N contains all the nurses in the hospital, the complement (\(N^{c}\)) would include all employees who are NOT nurses. This will include doctors, administrators, and any other staff members who are not nurses, regardless of their gender.

Key Concepts

Complement of a Set

The complement of a set is an important concept in set theory. It involves identifying all the elements that do not belong to a given set within a universal set. To define this, consider a universal set \(U\) and a subset \(A\) within \(U\). The complement of \(A\), denoted as \(A^c\), includes all elements in \(U\) that are not in \(A\).
Here’s how it works:

• Start with the universal set \(U\), which contains all possible elements.
• Take the subset \(A\), a specific group of elements from \(U\).
• The complement \(A^c\) consists of all the elements from \(U\) that are not in \(A\).

For example, in the context of a hospital, if set \(D\) represents all doctors, the set \(D^c\) would include nurses, administrators, and any other non-doctor personnel. Understanding the complement set helps highlight what is not included in the original group, broadening our understanding of the universal set.

Universal Set

In set theory, the universal set is a fundamental concept that encompasses all elements under consideration for a particular discussion or analysis.
When discussing sets and subsets, the universal set is the 'big picture' that contains every possible element that could be relevant.

• Think of it as a complete collection of items, such as all the employees in a hospital.
• Subsets are smaller groups, like nurses or doctors, formed from the universal set.
• Every other subset or complement we talk about is derived from this universal set.

By establishing a universal set \(U\), which might include all hospital employees, we ensure that every subset and complement makes logical sense within a defined boundary.
This overarching collection provides the context in which set operations, such as finding complements, are performed.

Mathematical Logic

Mathematical logic is the backbone of reasoning and problem-solving in mathematics, including set theory operations.
It provides the framework for understanding how and why operations are performed on sets. The central tenets are rooted in:

• Propositions: Statements that can either be true or false, important for defining sets clearly.
• Logical Operations: In the realm of sets, these include 'AND,' 'OR,' and 'NOT,' which relate closely to operations such as unions, intersections, and complements.
• Consistency: Logic ensures that set operations follow defined rules and produce reliable outcomes.

In the context of our original exercise, logical thinking helps us understand that a doctor, nurse, or administrator is defined based on their membership in specific sets.
Using logical operators, we can identify complements (such as \(D^c\) or \(N^c\)) to apply practical set operations efficiently. Understanding these logical principles is key to interpreting and manipulating sets accurately in various mathematical and real-world scenarios.