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Chapter 4: Problem 26

Find each indicated intersection or union. $$ \\{1,5,9\\} \cup\\{4,6,8\\} $$

Short Answer

Expert verified
\( \{1, 4, 5, 6, 8, 9\} \)

Step by step solution

01

- Understand the Symbols

Recognize that the symbol \(\bigcup\) represents the union of two sets.
02

- Identify the Elements in Each Set

The first set is \( \{1,5,9\} \) and the second set is \( \{4,6,8\} \).
03

- Combine the Sets

The union of two sets includes all unique elements from each set.
04

- Listing All Unique Elements

From \( \{1, 5, 9\} \) and \( \{4, 6, 8\} \), list all elements: \( \{1, 4, 5, 6, 8, 9\} \).

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Key Concepts

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

set theory
In mathematics, set theory is the study of sets, which are collections of abstract objects. A set can be anything: numbers, letters, symbols, or even other sets.
In set theory, objects in a set are called elements. For instance, in the set \( \{1,2,3\} \), \{1,2,3\} are the elements.
Sets are usually denoted by curly brackets \{ \}. Sets can be finite (with a limited number of elements) or infinite. They are fundamental in mathematics as they provide a foundation for a variety of other mathematical concepts.
union operation
The union operation is one of the basic operations on sets. When you form the union of two sets, you combine all the elements from each set into a new set without repeating any element.
The symbol for union is \( \cup \). For example, if we have two sets, \( \{a, b, c\}\footnotemark\{1, 5, 9\} \), performing the union would combine these sets into one, resulting in \( \{a, b, c, d, e, f\} \).
This shows how each unique element is included once in the resultant set, no matter how many times it appears in the original sets.
identifying unique elements
When performing the union of sets, it's important to only include unique elements. This means each element should appear only once in the resulting set.
Let's look at the given sets: \( \{1,5,9\} \) and \( \{4,6,8\} \).
None of the elements repeat between these two sets. Thus, their union would simply list all elements once: \( \{1, 4, 5, 6, 8, 9\} \).
If an element appears in both sets, it will still be listed only once in the union. This practice helps to keep the set representation concise and accurate.

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