Problem 23 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the smallest possible set (i.e., the set with the least number of elements) that contains the given sets as subsets. $$ \text { \\{Jill, John, Jack\\}, \\{Susan, Sharon\\} } $$

Step-by-Step Solution

Answer

The smallest possible set that contains the given sets as subsets is \(\{Jill, John, Jack, Susan, Sharon\}\).

Step 1: Identify the elements in both sets

The first set contains the following elements: Jill, John, and Jack. The second set contains the following elements: Susan and Sharon.

Step 2: Combine the elements of both sets into a single set

Now we need to combine the elements of both sets into a single set without repeating the elements. This new set should have all the elements of the two given sets. So we can write it as: \[ \text{Set 1: } \{Jill, John, Jack\} \] \[ \text{Set 2: } \{Susan, Sharon\} \]

Step 3: Find the smallest set that contains both given sets

To find the smallest set that contains both given sets, we simply combine the elements of Set 1 and Set 2, without repeating any element. This will give us the smallest possible set containing both given sets: \[ \text{Smallest Set: } \{Jill, John, Jack, Susan, Sharon\} \] Therefore, the smallest possible set that contains the given sets as subsets is the set \(\{Jill, John, Jack, Susan, Sharon\}\).

Key Concepts

Subsets

In set theory, a subset is a set that contains only elements from another set. When one set is entirely contained within another set, we call the first set a subset of the second. For example, if we have a set \(A = \{a, b, c\}\) and another set \(B = \{a, b\}\), then \(B\) is a subset of \(A\) because all of its elements \(\{a, b\}\) are also in \(A\). Identifying subsets is important in problems where we look to combine sets, as it helps us understand relationships and shared elements between sets. In the exercise, we are tasked with finding a set that has two other sets as its subsets, ensuring all elements are included in the larger set. This is the essence of constructing a union of sets while maintaining them as subsets of a larger set.

Union of Sets

The union of sets is a fundamental operation in set theory where we combine all elements from multiple sets into one. This method unites elements without duplicating them. When performing the union of sets \(A\) and \(B\), the result is a new set \(C\), which contains all elements of \(A\) and all elements of \(B\).For instance, consider two sets \(A = \{1, 2, 3\}\) and \(B = \{3, 4, 5\}\). When we find their union, we get \(A \cup B = \{1, 2, 3, 4, 5\}\). Notice that \(3\) is not repeated.In the given exercise, we had to find the smallest possible set containing the provided sets as subsets. So, we performed the union of \{Jill, John, Jack\} and \{Susan, Sharon\} to create the smallest set with all listed elements, resulting in \{Jill, John, Jack, Susan, Sharon\}. It is crucial to avoid duplication of elements while forming such unions.

Elements of a Set

In set theory, elements are the individual objects that belong to a set. They can be names, numbers, letters, or any defined objects. The importance of identifying elements stems from the need to understand what makes up a set and how these elements relate to subsets or in union operations.A set is often represented using curly brackets \(\{\}\) and elements are placed inside these brackets, separated by commas. For example, in a set \(C = \{apple, banana, cherry\}\), the elements are 'apple', 'banana', and 'cherry'.Considering the exercise, knowing the distinct elements within each set \{Jill, John, Jack\} and \{Susan, Sharon\} was the first step in achieving the union of these sets. It's essential that elements are clearly identified to ensure the correct composition of resulting sets. This clarity aids in tasks that involve checking subsets and performing unions, as well as maintaining the integrity of set operations.