Problem 29 In Exercises $29-34,$ find the... [FREE SOLUTION]

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Chapter 0: Problem 29

In Exercises $29-34,$ find the union of the sets. $$\\{1,2,3,4\\} \cup\\{2,4,5\\}$$

Short Answer

Expert verified

The union of the given sets is the set $\{1, 2, 3, 4, 5\}$.

Step by step solution

Identify the Individual Elements of Each Set

The first set contains the elements: $1, 2, 3, 4$ and the second set contains the elements: $2, 4, 5$.

Combine All Unique Elements

Merge the two sets together, while ensuring each element only appears once. As the numbers $2$ and $4$ appear in both sets, they do not need to be repeated.

Write Out the Union of the Sets

The union of the sets is a new set containing all unique elements from both sets. In this case, the union set includes the numbers $1, 2, 3, 4, 5$.

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Key Concepts

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

Sets

Sets are fundamental building blocks in mathematics and are essentially a collection of distinct objects or elements. These objects can be anything, such as numbers, letters, or even other sets. The key characteristic of a set is that it does not allow duplicate elements.
When representing sets, we typically use curly braces \{ \} to enclose the elements. For example, in the exercise, we have two sets: $\{1, 2, 3, 4\}$ and $\{2, 4, 5\}$. Each number within the braces is considered an element of the set.
Sets are often used to simplify complex problems by categorizing elements and understanding relationships between different collections.

Unique Elements

Unique elements refer to the distinct items that make up a set. In any given set, each element can appear only once, ensuring that there are no duplicates.
When we find the union of two sets, our goal is to gather all the distinct elements from both sets without repetition. For instance, in the given exercise, analyzing both sets $\{1, 2, 3, 4\}$ and $\{2, 4, 5\}$, we can see that numbers like $2$ and $4$ appear in both sets.

Number $2$ is unique within its own set, but it appears in both.
Number $4$ also appears in each set.
Remaining numbers $1, 3,$ and $5$ are unique across both sets, as they appear in only one set.

Thus, identifying these unique elements helps in forming what we call the union of the sets.

Merge Sets

Merging sets involves combining the elements of two or more sets while preserving the uniqueness of each element. This process is known as finding the union of sets.
In the exercise, when merging the sets $\{1, 2, 3, 4\}$ and $\{2, 4, 5\}$, we start by listing out elements of both sets but ensure not to repeat common elements more than once.
The process follows these steps:

List all the elements from the first set: $1, 2, 3, 4$.
Add elements from the second set: $2, 4, 5$, but only those which are not listed already in our first set's elements.

As a result, the final merged set, or union, is $\{1, 2, 3, 4, 5\}$, containing all elements without repeating any. This merging process helps in understanding how different sets can be combined in a way that maintains a simple and organized structure.

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91影视

In Exercises \(29-34,\) find the union of the sets. $$\\{1,2,3,4\\} \cup\\{2,4,5\\}$$

Short Answer

Step by step solution

Identify the Individual Elements of Each Set

Combine All Unique Elements

Write Out the Union of the Sets

Key Concepts

Sets

Unique Elements

Merge Sets

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