Problem 48 - 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 or are lawyers b. The set of all senators who are not Democrats or are lawyers

Step-by-Step Solution

Answer

a. \((D \cap F) \cup (D \cap L)\) b. \((U \setminus D) \cup L\)

Step 1: Part a: Determine the set of all Democrats who are female or are lawyers

First, let's determine the set of all Democrats who are female. This is the intersection of set D (Democrats) and set F (Females): \(D \cap F\). Similarly, let's determine the set of all Democrats who are lawyers. This is the intersection of set D (Democrats) and set L (Lawyers): \(D \cap L\). Now, we need to find the set of all Democrats who are either female or lawyers. This can be done using the union of the two sets we found above: \((D \cap F) \cup (D \cap L)\).

Step 2: Part b: Determine the set of all senators who are not Democrats or are lawyers

First, let's find the set of all senators who are not Democrats. This can be done using the complement operation: \(U \setminus D\). Now, we need to find the set of all senators who are either not Democrats or are lawyers. This can be done using the union operation on the set found above and the set L (Lawyers): \((U \setminus D) \cup L\).

Key Concepts

Intersection of Sets

The intersection of sets refers to the common elements that two or more sets share. When visualized using Venn diagrams, the intersection is the area where the circles representing these sets overlap. Mathematically, if we have two sets, say A and B, their intersection is written as \( A \cap B \). It includes all elements that A and B have in common.

For the provided example, where Democrats (D) and females (F) are considered, the intersection \( D \cap F \) contains all individuals who are both Democrats and females. Similarly, the intersection \( D \cap L \), where L represents lawyers, includes all Democrats who are also lawyers. The concept of set intersection is foundational in set theory and provides a way to identify shared characteristics between groups.

Union of Sets

The union of sets is another fundamental concept in set theory. It represents the combination of all elements found in either of the sets, without duplication. If we denote two sets by A and B, their union is denoted by \( A \cup B \), and it will include any element that is in A, in B, or in both.

In our exercise, we used the union concept to find all Democrats who are either female or lawyers. By denoting the set of Democrats and females as \( D \cap F \) and the set of Democrats and lawyers as \( D \cap L \), their union \( (D \cap F) \cup (D \cap L) \) covers all individuals who are either female Democrats, lawyer Democrats, or both, illustrating how the union operation brings together different groups under a single set.

Complement of Sets

The complement of a set consists of all elements that are not in the given set but are present in a universal set. Generally, if U is the universal set and A is a subset of U, the complement of A is denoted by \( U \setminus A \) or sometimes by \( A' \). The complement operation effectively inverts the membership of elements regarding set A.

In the context of our problem, the set of all senators who are not Democrats — written as \( U \setminus D \) — is the complement of the set of Democrats within the universal set U of all senators. This is a powerful tool in set theory to express exclusion, and when combined with other set operations, it allows for the construction of complex groupings based on inclusive and exclusive criteria.

Applied Mathematics

Applied mathematics involves the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Set theory, with concepts like intersection, union, and complement, is extensively used in applied mathematics for solving real-world problems. These problems can range from computer science algorithms that filter and merge data to sociological studies categorizing diverse groups of people.

In our case, we use set theory to describe and analyze groups of senators based on their political affiliation, gender, and profession. This simple exercise exemplifies how mathematical concepts are not just theoretical constructs but have real-life implications and applications, providing structured ways to organize, analyze, and visualize complex information.