Problem 33 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the concept

The union of two sets, represented typically by the symbol \( \cup \), is a set of all unique elements found in either set or in both. It's like collecting all distinct elements from the two sets.

Step 2: Union of set with an empty set

When finding the union of a set, say \( A \), with an empty set (denoted as \( \varnothing \)), the result will simply be the original set \( A \) again. This happens because an empty set doesn't contribute any elements to change the original set.

Step 3: Applying the concept to the exercise

Since the empty set doesn't contribute any elements, the union of \( \{ a, e, i, o, u\} \) and \( \varnothing \) is the set \( \{ a, e, i, o, u\} \) itself.

Key Concepts

Set Theory

Set theory is a fundamental concept in mathematics that involves the study of sets, which are collections of distinct objects. These objects, often called elements, can be anything: numbers, letters, symbols, or even other sets.

• Notation: Sets are usually denoted by capital letters like \( A \), \( B \), \( C \), and so on, while their elements are listed within curly braces such as \( \{1, 2, 3\} \).
• Operations: In set theory, operations such as union, intersection, and difference are used to combine or compare sets.

These operations are crucial for establishing relationships between sets. In this context, the union operation, denoted by the symbol \( \cup \), represents a key operation that combines two sets into a new set containing all distinct elements from both.

Empty Set

The concept of an empty set is a fundamental piece of set theory. An empty set, denoted by the symbol \( \varnothing \), is a set that contains no elements.

• Uniqueness: It is unique, meaning there is only one empty set regardless of what kind of elements are considered.
• Union with Other Sets: When you take the union of any set with an empty set, the result is the original set. This is because the empty set adds no elements to the union.
• Role in Mathematics: The empty set serves as the identity element for the union operation, much like zero in addition or one in multiplication.

Understanding the empty set helps clarify results like \( \{ a, e, i, o, u\} \cup \varnothing = \{ a, e, i, o, u\} \), where the presence of an empty set leaves the original set unchanged.

Unique Elements

One of the key properties in set operations is the inclusion of only unique elements. This means that no element appears more than once in a set.

• Union of Sets: The union operation exemplifies this property well. When you perform the union of two sets, you gather all elements from both sets but do not repeat any element.
• Example: Consider sets \( \{a, b, c\} \cup \{a, c, d, e\} \). The union will be \( \{a, b, c, d, e\} \), with each element appearing only once.

This concept ensures that the results of set operations are precise and predictable. Thus, in the exercise, the union \( \{ a, e, i, o, u\} \cup \varnothing\) retains only the unique elements of the set \( \{ a, e, i, o, u\}\), because the empty set contributes no additional elements.