Problem 47 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(U\) denote the set of all senators in Congress and let $$ \begin{array}{l} D=\\{x \in U \mid x \text { is a Democrat }\\} \\ R=\\{x \in U \mid x \text { is a Republican\\} } \\ F=\\{x \in U \mid x \text { is a female }\\} \\ L=\\{x \in U \mid x \text { is a lawyer }\\} \end{array} $$ Write the set that represents each statement. a. The set of all Democrats who are female b. The set of all Republicans who are male and are not lawyers

Step-by-Step Solution

Answer

a. \( D \cap F = \{x \in U \mid x \text{ is a Democrat and x is a female} \} \) b. \( (R \cap \bar{F}) \cap \bar{L} = \{x \in U \mid x \text{ is a Republican, x is a male, and x is not a lawyer} \} \)

Step 1: a. The set of all Democrats who are female

To find the set of all Democrats who are female, we have to find the intersection of set \(D\) and set \(F\). This can be written as: \( D \cap F = \{x \in U \mid x \text{ is a Democrat and x is a female} \} \) This represents all senators who are both in the \(D\) and \(F\) sets.

Step 2: b. The set of all Republicans who are male and are not lawyers

For this statement, we need to find the set of all Republicans who are male and are not lawyers. We know that Republicans are represented by set \(R\) and lawyers by set \(L\). First, we find the set of all male senators by taking the complement of the female senators' set. So, male senators are represented by the set \(\bar{F}\). Now, we need to find the intersection of set \(R\) and set \(\bar{F}\) to find all Republicans who are male: \( R \cap \bar{F} = \{x \in U \mid x \text{ is a Republican and x is a male } \} \) Next, we need to find all senators who are not lawyers. We find the complement of the set \(L\), which is \(\bar{L}\). Finally, we find the intersection of the set \(R \cap \bar{F}\) and the set \(\bar{L}\) to get the set of all Republicans who are male and are not lawyers: \( (R \cap \bar{F}) \cap \bar{L} = \{x \in U \mid x \text{ is a Republican, x is a male, and x is not a lawyer} \} \) This represents all senators who fulfill the given conditions in statement b.

Key Concepts

Set Intersection

In set theory, the intersection of sets is a fundamental concept used to find common elements. For instance, if you have two sets, say set \(A\) and set \(B\), the intersection of these sets, denoted as \(A \cap B\), is the set of all elements that belong to both \(A\) and \(B\).
To visualize, imagine two circles overlapping in some regions. The overlapping part represents the intersection, holding the common elements in both circles.

In the context of the given exercise, to find Democrats who are female, the problem requires finding the intersection of sets \(D\) (Democrats) and \(F\) (Females), which is written as \(D \cap F = \{x \in U \mid x \text{ is a Democrat and x is a female}\}\). This operation helps filter senators who satisfy both conditions.

Set Complement

The set complement is another essential concept in set theory. When you have a set \(A\), the complement, denoted as \(\bar{A}\) or sometimes \(A^c\), includes all elements not in \(A\), provided they belong to the universal set \(U\).
This is especially useful when you need to find elements that satisfy a condition that negates an initial one.

• Example: If \(F\) is the set of all female senators, \(\bar{F}\) will be the set of all male senators.

In our exercise, we utilized the set complement to find male senators and those who are not lawyers. We applied the complement concept with sets like \(F\) for females (giving us males) and \(L\) for lawyers, emphasizing how the complement is crucial in defining what is "outside" a particular set.

Problem Solving

Effective problem-solving in mathematics often involves applying set operations like intersection and complement. These operations enable the breakdown of complex problems into simpler, manageable parts. The key steps in problem-solving with sets include:

• Identifying the relevant sets from the given information.
• Deciding which operations are needed (e.g., intersection, union, complement).
• Performing these operations in logical order.

In the exercise, we tackled finding Democrats who are female by intersecting sets and Republicans who are male but not lawyers through a series of intersections and complements. Following a systematic approach helps maintain clarity and ensures that all conditions are met accurately.

Logic in Mathematics

Mathematical logic provides the foundation for reasoning and solving problems systematically. It is integral to set theory, where statements and operations reflect logical deductions. The principles of logic guide how we achieve a solution and ascertain the truth of mathematical expressions.

Consider the logical steps in the given exercise. We start with basic sets defining political affiliation, gender, and profession. Then, we apply logical operations (intersection and complement) to draw conclusions about specific subsets of individuals in Congress.

• Understanding logical operators like "and" and "not" helps refine which elements meet the set conditions.
• Logic in this problem assures precise outcomes when combining different attributes, such as gender and profession.

By internalizing logical processes, complex tasks become straightforward, leading to reliable and valid solutions to mathematical problems.