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Chapter 0: Problem 33

Find the union of the sets. $$\\{a, e, i, o, u\\} \cup \varnothing$$

Short Answer

Expert verified
The union of the sets \( \{a, e, i, o, u\} \) and \( \varnothing \) is \( \{a, e, i, o, u\} \)

Step by step solution

01

Understand the Sets

We have two sets here: The first set contains single letter vowels of the English alphabet \( \{a, e, i, o, u\} \), and the second set is an empty set, denoted by \( \varnothing \).
02

Perform Union Operation

The union of the sets is the set of elements which are in either set and in both. In this case, when we add (union) an empty set with a non-empty set, we get the non-empty set. So, \( \{a, e, i, o, u\} \cup \varnothing = \{a, e, i, o, u\} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sets in Mathematics
In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. Objects can be anything: numbers, people, letters, etc., and they are called the elements of the set. Sets are typically enclosed in curly braces \(\{\}\) and are fundamental to various mathematical disciplines.

To grasp the nature of sets, it's crucial to know certain properties. Sets are defined purely by their elements, not by the order the elements are listed or by any repetition of elements. For example, the set \(\{a, e, i, o, u\}\) is a collection of the vowels in the English alphabet with each vowel appearing only once, regardless of their order in the list.
Empty Set
The empty set, denoted by \(\varnothing\) or \(\{\}\), is the unique set having no elements. It's a fundamental concept in mathematics as it serves as a foundation for building more complex set theories. Despite having no elements, the empty set is still a set, following the property that it is uniquely defined by its elements鈥攐r, in this case, the lack thereof.

The empty set is an integral part of many mathematical operations and concepts. In our example, \(\varnothing\) is combined with a set of vowels, showcasing how it integrates with other sets.
Union Operation
The union of two or more sets is a basic operation in set theory that combines all distinct elements from each set into a new set. The union operation is denoted by the symbol \(\cup\). When sets are joined using the union, any overlapping elements are included only once in the result.

For instance, if Set A is \(\{1, 2, 3\}\) and Set B is \(\{3, 4, 5\}\), their union \(A \cup B\) is \(\{1, 2, 3, 4, 5\}\), merging all unique elements from both sets. In the case of our example, the union of \(\{a, e, i, o, u\}\) and \(\varnothing\) results in \(\{a, e, i, o, u\}\), as the empty set contributes no new elements.
Basic Set Theory
Basic set theory lays the groundwork for understanding and working with sets in mathematics. It covers elementary concepts such as what sets are, along with the operations that can be performed on them such as union, intersection, and set difference.

It also delves into more complex notions like subsets, power sets, and Cartesian products. These foundational ideas build upon each other to form the basis of more abstract mathematical theories. Being familiar with basic set theory is essential for any mathematical endeavor as it is widely used in different areas of mathematics, including algebra, geometry, and analysis.

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