In Exercises 33–38, find the union of the sets. { a, e, i, o, u } ⋃ ∅

Verified step by step guidance

1. Identify the sets involved in the union operation: Set A = \{ a, e, i, o, u \} and Set B = \emptyset (the empty set).
2. Recall the definition of the union of two sets: The union of two sets A and B, denoted as A \cup B, is the set of elements that are in A, in B, or in both.
3. Consider the properties of the empty set: The empty set, \emptyset, contains no elements.
4. Apply the union operation: Since the empty set has no elements, the union of any set with the empty set is simply the original set.
5. Conclude that the union of \{ a, e, i, o, u \} and \emptyset is \{ a, e, i, o, u \}.

Key Concepts

Sets

A set is a well-defined collection of distinct objects, considered as an object in its own right. In this context, sets can contain elements such as numbers, letters, or other sets. The notation { a, e, i, o, u } represents a set of vowels, while ∅ denotes the empty set, which contains no elements.

Union of Sets

The union of two sets is a new set that contains all the elements from both sets, without duplication. It is denoted by the symbol '∪'. For example, if we take the union of { a, e, i, o, u } and ∅, the resulting set will include all the elements from the first set, as the empty set contributes no additional elements.

Empty Set

The empty set, denoted as ∅, is a unique set that contains no elements. It serves as the identity element for the union operation, meaning that the union of any set with the empty set will yield the original set. Understanding the role of the empty set is crucial when performing operations involving unions.