Problem 49 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(U\) denote the set of all students in the business college of a certain university. Let \(A=\\{x \in U \mid x\) had taken a course in accounting \(B=\\{x \in U \mid x\) had taken a course in economics\\} \(C=\\{x \in U \mid x\) had taken a course in marketing \(\\}\) Write the set that represents each statement. a. The set of students who have not had a course in economics b. The set of students who have had courses in accounting and economics c. The set of students who have had courses in accounting and economics but not marketing

Step-by-Step Solution

Answer

a. The set of students who have not had a course in economics can be represented as \(B^c = U \setminus B\). b. The set of students who have had courses in accounting and economics can be represented as \(A \cap B\). c. The set of students who have had courses in accounting and economics but not marketing can be represented as \((A \cap B) \setminus C\).

Step 1: Find the complement of set B

Determine the complement of set B, which represents students who haven't taken an economics course in relation to the universal set U. The complement of set B is given by \(B^c = U \setminus B\). b. The set of students who have had courses in accounting and economics: To find this set, we need to determine the intersection of set A (students who have taken an accounting course) and set B (students who have taken an economics course).

Step 2: Find the intersection of set A and set B

Determine the intersection of sets A and B, which represents students who have taken both accounting and economics courses. The intersection of sets A and B is given by \(A \cap B\). c. The set of students who have had courses in accounting and economics but not marketing: To find this set, we need to determine the intersection of set A (students who have taken an accounting course) and set B (students who have taken an economics course) excluding students who are part of set C (students who have taken a marketing course).

Step 3: Find the intersection of set A and set B excluding set C

Determine the intersection of sets A and B, and exclude the students who have taken a marketing course (students in set C). The desired set can be found using the expression \((A \cap B) \setminus C\). This expression represents students who have taken both accounting and economics courses (A ∩ B), but have not taken a marketing course (excluding students in set C).

Key Concepts

Intersection of Sets

In set theory, the intersection of two sets includes all elements that are common to both sets. This shared space allows us to understand where two groups overlap. When dealing with the intersection of sets, we use the symbol \( \cap \).
For example, if we consider sets \(A\) and \(B\), the intersection \( A \cap B \) includes all elements that appear in both sets.
In our problem scenario, set \(A\) consists of students who have taken an accounting course, while set \(B\) comprises students who have taken an economics course. Thus, the intersection \( A \cap B \) describes students who have taken both courses — accounting and economics.
This concept is straightforward and helps us identify individuals or objects that belong to multiple categories simultaneously.

Complement of a Set

The complement of a set contains all elements that are not part of the specified set, but still part of the universal set \(U\). Essentially, it represents the "opposite" or "left-out" group. We represent the complement of a set \(B\) by \( B^c \).
In the exercise, set \(B\) includes students who took an economics course. Thus, its complement \( B^c \) includes all students within the business college who did not take economics.
Finding complements can be extremely useful, especially when you want to identify what is missing or what does not belong to a particular subset in a given universal context.

Set Operations

Set operations are techniques we use to combine, compare, or modify sets. These operations form the basis of set theory and include intersection, union, and complement, among others.
In our exercise, various set operations come into play. For instance, the intersection operation allows us to find students enrolled in both accounting and economics. We denote this by \( A \cap B \).

• Intersection: Finds common elements between sets. Notation: \( \cap \).
• Complement: Identifies elements not in a set within the universal set. Notation: \( B^c \).
• Set Difference: Shows elements in one set but not another. For \((A \cap B) \setminus C\), it identifies students who took accounting and economics, but not marketing.

Understanding and using set operations enhance our ability to solve complex problems by breaking them down into manageable components.