/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Find the union of the sets. \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 31

Find the union of the sets. \(\\{1,3,5,7\\} \cup\\{2,4,6,8,10\\}\)

Short Answer

Expert verified
The union of the sets \{1,3,5,7\} and \{2,4,6,8,10\} is \{1,2,3,4,5,6,7,8,10\}.

Step by step solution

01

Understanding the Union of Sets

In set theory, the union of two sets is a new set containing all distinct elements from both the sets. If an element belongs to either of the given sets (or to both), it will belong to the union set. This is represented by the 'cup' symbol, \(\cup\).
02

Identifying the Elements in each Set

Identify the unique elements in each set. The first set has the elements \{1,3,5,7\} and the second set has the elements \{2,4,6,8,10\}.
03

Merging the Sets

As we look at these sets, we can see that they already have distinct elements. So, for the union operation, we simply combine the elements from both sets. Merging these sets, we get \{1,2,3,4,5,6,7,8,10\}.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Set Theory
Set theory is a fundamental part of mathematics, dealing with the collection of objects, known as sets. It allows us to understand how these collections interact, overlap, and combine with each other. Set theory provides us with the tools to make more sense of how we group numbers or objects into categories. For example, when you have a group of even numbers and a group of odd numbers, set theory helps you manage these groups and find interesting properties about them, such as calculating their union or intersecting them. Understanding sets and how they work is crucial to many areas of mathematics and science.
Distinct Elements
Distinct elements refer to unique items in a set. In set theory, a set is defined as a collection of well-defined, distinct objects. This means each element in a set is different from others and usually doesn鈥檛 repeat. For instance, in the set \(\{1, 3, 5, 7\}\), each of the numbers is unique to that set.
  • Sets only include distinct items, so duplicates are ignored automatically.
  • This uniqueness concept is essential when combining sets, like in union operations, to avoid repetition.
Recognizing distinct elements helps when identifying what makes up a particular set and in understanding operations like unions and intersections.
Symbol for Union
The symbol for union in set theory is a cup shape, represented by \(\cup\). This symbol is used to denote the operation of union between two sets.
  • Whenever you see \(A \cup B\), it means you're taking all elements from set \(A\) and set \(B\) to form a new set.
  • Union helps in combining different groups while maintaining only one instance of each distinct element.
The use of \(\cup\) is straightforward: you list out all unique elements from both sets. It鈥檚 a fundamental operation that visualizes how two or more sets can be merged into one larger set containing all of their elements.
Merging Sets
Merging sets is the process of combining elements from two or more sets into a single set. In the context of set theory, this operation is typically defined by the union of sets. When you merge sets:
  • You include every distinct element found in any of the sets being merged.
  • If there are common elements between the sets, they will appear only once in the new set, preventing duplicates.
For example, when given the sets \(\{1, 3, 5, 7\}\) and \(\{2, 4, 6, 8, 10\}\), merging them involves listing every distinct element once, resulting in \(\{1, 2, 3, 4, 5, 6, 7, 8, 10\}\). Merging sets through union is a powerful way to understand how collections relate and interact in mathematics.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Explain how to convert from decimal to scientific notation and give an example.

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$\text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. }$$ How old is the son? b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The trinomial \(x^{2}-4 x-4\) is a prime polynomial.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. First factoring out the greatest common factor makes it easier for me to determine how to factor the remaining factor, assuming that it is not prime.

Perform the indicated operations. $$ [(7 x+5)+4 y][(7 x+5)-4 y] $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.