Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(U\) denote the set of all employees at Universal Life Insurance Company and let $$ \begin{array}{l} \boldsymbol{T}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { drinks tea }\\} \\ \boldsymbol{C}=\\{\boldsymbol{x} \in \boldsymbol{U} \mid \boldsymbol{x} \text { drinks coffee }\\} \end{array} $$ Describe each set in words. a. \(T^{c}\) b. \(C^{c}\)

Step-by-Step Solution

Answer

a. The complement of the set \(T\) (denoted by \(T^{c}\)) represents all employees who do not drink tea. b. The complement of the set \(C\) (denoted by \(C^{c}\)) represents all employees who do not drink coffee.

Step 1: a. Tᶜ: Complement of tea drinkers

The complement of the set \(T\) (denoted by \(T^{c}\)) represents all employees who do not drink tea. In other words, if an employee is not in the set \(T\), then they are in the set \(T^{c}\). #b. Describing the complement of the set C#

Step 2: b. Cᶜ: Complement of coffee drinkers

The complement of the set \(C\) (denoted by \(C^{c}\)) represents all employees who do not drink coffee. In other words, if an employee is not in the set \(C\), then they are in the set \(C^{c}\).

Key Concepts

Universal sets

Universal sets play a foundational role in set theory. A universal set, often denoted by the letter "U", is a set that contains all the objects or elements relevant to a particular discussion or problem. In our example, the universal set is the set of all employees at Universal Life Insurance Company. This means that every employee, regardless of whether they drink tea, coffee, or neither, is part of this universal set.

Understanding the universal set helps in understanding the limits or boundaries of what we are discussing. When we describe subsets, such as the set of tea drinkers or coffee drinkers, we are selecting specific groups from this universal set. It's important to remember that the universal set can vary depending on the context of the problem. Another context might define a different group of elements entirely, adjusting the basis for all subsets within.

• Universal sets contain all elements of discussion.
• Defined by the context of the problem.
• Sets examples in context: all employees at a company.

Set complements

In set theory, a set complement is a way of defining elements that do not belong to a particular subset of the universal set. When talking about the complement of a set, we refer to those elements in the universal set that are not in the subset. This is denoted with a superscript "c". For instance, the complement of the set of tea drinkers, noted as \(T^{c}\), includes all employees who do not drink tea.

Similarly, the complement of the set of coffee drinkers, \(C^{c}\), would include all employees who do not drink coffee. Complements help us understand not just what is within a set, but just as importantly, what is outside it. It gives us insights into what remains when certain characteristics are removed or negated. Complements are closely related to the idea that every set paired with its complement forms the whole of the universal set without any overlap.

• Complement sets show elements not in a given subset.
• Denoted with a "c" superscript (e.g., \(T^{c}\)).
• Both set and its complement together complete the universal set.

Mathematical notation

Set theory heavily relies on mathematical notation for clear and concise communication. Using symbols and structured expressions, we can convey complex ideas with ease. In the context of sets, notation like \(\{x \in A \mid \, \text{condition}\}\) is used to concise elements selection, meaning the set of all elements "x" in set "A" that meet the condition described.

The concepts of complements within set theory utilize the notation of \(A^{c}\), which represents all elements in the universal set that are not in subset "A". Proper notation helps prevent misunderstandings and errors as it provides a universal language understood by mathematicians worldwide. This allows for sharing and cooperation across different cultures and disciplines.

• Mathematical notation allows concise expression of ideas.
• Common symbols include union, intersection, and complements.
• Ensures clarity and precision in mathematical communication.