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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.

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.

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.