Problem 40 - 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. \(T \cup C\) b. \(T \cap C\)

Step-by-Step Solution

Answer

a. The set \(T \cup C\) includes "all employees who drink tea or coffee or both". b. The set \(T \cap C\) includes "all employees who drink both tea and coffee".

Step 1: a. Describing the set T ∪ C

The union T ∪ C consists of all elements that are in either T or C or both. This means we are looking for all employees who either drink tea or coffee or both. In words, the set T ∪ C includes "all employees who drink tea or coffee or both".

Step 2: b. Describing the set T ∩ C

The intersection T ∩ C consists of all elements that are common to both sets T and C. This means we are looking for the employees who drink both tea and coffee. In words, the set T ∩ C includes "all employees who drink both tea and coffee".

Key Concepts

Union of Sets

A union of sets is a concept where we combine elements from multiple sets. Imagine you have two groups of people: one group drinks tea and the other drinks coffee. The union of these two groups will include anyone who drinks either tea or coffee or both. Think of the union as a way to cast a wide net to gather all possible members without duplicating individuals.

In the formal set notation, if you have two sets, say \(A\) and \(B\), their union is represented by \(A \cup B\). The symbol \(\cup\) is read as "union."

For example:

• Set \(A = \{1, 2, 3\}\)
• Set \(B = \{3, 4, 5\}\)

The union of these sets, \(A \cup B\), will be \(\{1, 2, 3, 4, 5\}\). Notice that 3 is not duplicated even though it appears in both sets.

You often use unions when you want to ensure that you encompass all options available from different groups or categories.

Intersection of Sets

The intersection of sets is where we find common elements between two or more sets. Imagine two groups: one drinks tea, and another drinks coffee. The intersection is the group of employees who enjoy both beverages.

When we talk about the intersection, we're looking for shared traits. So, in set theory, intersection focuses purely on the overlap.

In the language of set theory, if you have two sets \(A\) and \(B\), their intersection is denoted by \(A \cap B\). The symbol \(\cap\) stands for "intersection."

For example:

• Set \(A = \{1, 2, 3\}\)
• Set \(B = \{3, 4, 5\}\)

The intersection \(A \cap B\) results in \(\{3\}\), which is the only number common to both sets.

Intersections are useful when you need to find shared characteristics or mutual interests between groups.

Universal Set

In set theory, the universal set is a larger set that contains all the objects or elements under consideration for a particular discussion. When we delve into smaller sets, these are often subsets of the universal set.

Picture the universal set as a giant umbrella covering all the elements relevant to a specific context. For example, if we are talking about employees at a company, the universal set would be the group containing all employees.

When defining or working with subsets, such as those who drink tea or coffee, these subsets are all contained within the universal set.

Identifying the universal set is important because it helps to anchor other discussions and subsets within a defined boundary. This ensures that all subsets and their interactions are always framed with a clear overall context in mind.

If you are working with any sets, always start by defining your universal set so that all subsequent work is rooted in this foundational group.