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Chapter 7: Problem 45

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. \(D \cap M^{c}\) b. \(D \cap A\)

Short Answer

Expert verified
a. \(D \cap M^{c}\) represents the set of all employees who are female doctors. b. \(D \cap A\) describes the set of all employees who are both doctors and administrators.

Step by step solution

01

Understand the given sets

First, let's understand the given sets: - \(N\): The set of employees who are nurses. - \(D\): The set of employees who are doctors. - \(A\): The set of employees who are administrators. - \(M\): The set of employees who are males. - \(F\): The set of employees who are females. Now let's analyze the two cases and describe the sets in words.
02

Analyze and describe \(D \cap M^{c}\)

Here, we have a set operation, \(D \cap M^{c}\). This represents the intersection of the set of doctors (\(D\)) and the complement of the set of males (\(M^{c}\)). In other words, we are looking for employees who are both doctors and NOT males. Since the only other option in this context is being female, this set represents female doctors. So, \(D \cap M^{c}\) describes the set of all employees who are female doctors.
03

Analyze and describe \(D \cap A\)

Next, we have the set operation \(D \cap A\). This represents the intersection of the set of doctors (\(D\)) and the set of administrators (\(A\)). In this case, we are looking for employees who are both doctors and administrators. This set represents employees who have a dual role: they are qualified as doctors and also take on administrative duties in the hospital. So, \(D \cap A\) describes the set of all employees who are both doctors and administrators.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Set Operations
In set theory, set operations allow us to combine, compare, and manipulate different sets to derive new sets. These operations help us analyze collections of objects that share common characteristics. Some basic operations include union, intersection, and complement.
  • Union (\( \igcup \) ): Combines all elements from two sets.
  • Intersection (\( \igcap \) ): Includes elements found in both sets.
  • Complement (\( ^c \) ): Contains elements not in the specified set, relative to a universal set.
Breaking down complex problems becomes easier by applying these operations to understand the relationships within sets. In practical scenarios, they help in organizing data, defining roles, and clarifying tasks within groups.
Intersection of Sets
The intersection of sets is a fundamental concept used to identify elements that are common to two or more sets. In symbolic terms, the intersection is denoted by \( \igcap \).
For example, if you have Set \( A \) and Set \( B \), their intersection, \( A \igcap B \), will contain only the elements present in both.

Example in Context:

Consider the problem exercise where \( D \igcap A \) is analyzed. Here, \( D \) represents doctors, and \( A \) represents administrators.
The intersection \( D \igcap A \) signifies employees who have dual responsibilities as doctors and administrators. Such intersections help in identifying shared roles or characteristics within groups, making it a useful tool for analysis and decision-making.
Complement of a Set
The complement of a set refers to the elements not included in a particular set but are part of a larger universal set. It is denoted by \( ^c \). When dealing with complements, you're essentially looking at everything outside the specified set within the universe of discourse.

Practical Example:

In the given exercise, \( M^c \) indicates employees who are not males. If \( M \) is the set of male employees, then \( M^c \) will include all non-male (or female) employees.
This concept helps in filtering information and narrowing down criteria to focus on specific groups or characteristics, which is essential for organization and resource management.

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