Back To Course

High School Precalculus: Tutoring Solution32 chapters | 265 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Log in here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*David Liano*

After completing this lesson, you will know how to write a set using words and using mathematical symbols. You will also know how to define sets and how to interpret sets that are displayed in set notation.

A **set** is simply a collection of items called elements, or members of the set. Each element is distinguishable from the other elements. Examples of sets are probably the best way to illustrate what a set is.

A set could be the countries of Europe. France would be an element of this set. The country of Argentina would not be an element of this set because it is located in South America. The city of Rome would not be an element of this set because it is a city in Europe, not a country in Europe. In the context of mathematics, a set could be all the integers greater than 10 and less than 20. The numbers 12 and 17 would be elements of this set, while the numbers 20 and 35 would not be elements of this set.

We can make any type of set we want. For instance, we could combine the sets above into one set. The elements in a set do not need to have any relationship except that they are elements of the same set. For instance, Babe Ruth and the number 1,000,000 could be in the same set. However, there is usually some connection between the elements of a set to make the set practical and useful.

Braces {} are usually used when writing down a set. It is often common to use capital letters to name a set. Let's say set A has the elements of 3, 5, and 7. We would write is as follows:

*A* = {3, 5, 7}

Set A has three elements. We can list the elements in any order, and we can list elements more than once. We can also write set A as follows:

*A* = {5, 3, 7}*A* = {3, 5, 7, 7}

However, the elements have not changed, and there are still only three elements. We can also have sets within sets. That means that sets can be elements of other sets. Let's look at the following example:

*B* = { *a*, {*a*}, *b*, *c*, {*d*, *e*} }

In set B, there are five elements. Two of the elements are sets of letters. The other three elements are single letters. Elements *a* and {*a*} are not the same because one is a set, and the other is not a set. In addition, the letters *d* and *e* are not elements of set B, but the set {*d*, *e*} is an element of set B. This distinction between elements and sets is straightforward but often is a difficult rule to apply.

For sets with many elements, it might be helpful to abbreviate using the ellipse (. . .) symbol. For instance, the set containing all natural numbers from 1 to 199 would be cumbersome to write out completely. We could abbreviate as follows:

*S* = {1, 2, 3, â€¦ 199}

To show that an element is part of a set, we use a curvy E symbol. The number 5 is an element in set S, and this is shown in Figure 1 using the curvy E symbol (below).

We use certain letters to define various number systems. This helps to better define sets and to make them easier to write. We will use the following capital letters for the respective number system sets:

** N** = Natural Numbers {1, 2, 3, . . .}

** Z** = Integers {â€¦ -3, -2, -1, 0, 1, 2, 3, . . .}

** Q** = Rational Numbers

** I** = Irrational Numbers

** R** = Real Numbers

Let's take the set we mentioned earlier of natural numbers from 1 to 199. Using the letter ** N** for natural numbers, we can write the set in set notation as shown in Figure 2 (below).

Interpret the vertical line after the variable *x* to mean 'such that.' Let's now write this set in words: set *A* consists of all elements *x* such that *x* is a natural number and *x* is less than 200.

Let's consider two sets *A* and *B* shown below:

*A* = {1, 3, 5}*B* = {1, 3, 5, 7, 9}

We can say that A is a **subset** of *'B* because all the elements of set *A* are also elements of set *B*. Set *A* is more specifically a **proper subset** of *B* because *A* does not equal *B*. In other words, there are some elements in set *B* that are not in set *A*. This can also be shown using a Venn diagram as shown in Figure 3 (below).

The **union of two sets** is the set of all elements that are members of one set or the other. Let's look again at two sets: *A* and *B*:

*A* = {1, 2, 3}*B* = {2, 3, 4}

We use a capital U to specify the union of two sets. So *A U *B* = {1, 2, 3, 4}. Notice that neither set is a subset of the other. *

*The intersection of two sets is the set of all elements that are members of both sets. We use an upside down U to specify the intersection of two sets. The intersection of sets *A

*We can also show the union and intersection of sets *A* and *B* using proper set notation as shown in Figure 5 (below). *

Let's cover one more thing about set notation. The **cardinality** of a set is the number of elements in a set. We will keep this part of the discussion within the category of finite sets. Let's say that we have the following set: *D* = {3, 7, 8, 15}. This set has four elements. Therefore, the cardinality of set *D* is 4. To show cardinality in symbols, we enclose the name of the set between two vertical lines:

A set with no elements is called the **empty set**. We can use the braces to show the empty set: { }. More commonly, this symbol, Ø, is used to show the empty set. The empty set has no members, so we can say that all elements of the empty set are elements of any other set. Does this definition sound familiar? Therefore, the empty set is a subset of any set.

Let's review. **Set notation** is used to help define the elements of a set. The symbols shown in this lesson are very appropriate in the realm of mathematics and in mathematical logic. When done properly, a set described in words or in symbols will clearly show all the elements of that set. When describing a set, we need to make sure that there will be no ambiguity to anyone reading the set.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Log in here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
4 in chapter 28 of the course:

Back To Course

High School Precalculus: Tutoring Solution32 chapters | 265 lessons

- SIE Exam Study Guide
- Indiana Real Estate Broker Exam Study Guide
- Grammar & Sentence Structure Lesson Plans
- Foundations of Science Lesson Plans
- Career, Life, & Technical Skills Lesson Plans
- Business Costs, Taxes & Inventory Valuations
- Using Math for Financial Analysis
- Assessments in Health Education Programs
- Governmental Health Regulations
- Understanding Health Education Programs
- AFOQT Prep Product Comparison
- ACT Prep Product Comparison
- CGAP Prep Product Comparison
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison

- What is Deadlock? - Definition, Examples & Avoidance
- El Hombre que se Convirtio en Perro: Author, Summary & Theme
- Achilles in The Iliad: Character Analysis & Description
- A Wrinkle in Time Chapter 5 Summary
- Roald Dahl Project Ideas
- Media Literacy Activities for High School
- Letter M Activities
- Quiz & Worksheet - Shang Dynasty Religion & Culture
- Quiz & Worksheet - Alternative Assessment Types
- Quiz & Worksheet - Population Composition
- Quiz & Worksheet - Minimalist Painters
- Analytical & Non-Euclidean Geometry Flashcards
- Flashcards - Measurement & Experimental Design
- Social Emotional Learning SEL Resources for Teachers
- NGSS | Next Generation Science Standards Guide for Teachers

- Executing a Business Impact Analysis
- MTLE Physical Education: Practice & Study Guide
- ILTS English Language Arts (207): Test Practice and Study Guide
- Principles of Marketing Syllabus Resource & Lesson Plans
- Holt McDougal Algebra 2: Online Textbook Help
- The Virginia Dynasty (1801-1825)
- Management Basics
- Quiz & Worksheet - Epithelial Tissue Diseases
- Quiz & Worksheet - The Wilderness Campaign
- Quiz & Worksheet - Function of Arteries
- Quiz & Worksheet - Scientific Theory
- Quiz & Worksheet - Prometaphase

- Warning Coloration in Animals: Examples, Overview
- Gallbladder: Definition, Function & Location
- Study.com's Workforce College Accelerator for Employees
- What is a Distance Learning Course?
- SBEC Technology Application Standards for Teachers
- Memoir Lesson Plan
- Texas Educators' Code of Ethics
- Illinois Science Standards for First Grade
- Study.com's GED Program for Enterprise
- 5th Grade Science Standards in California
- Minnesota State Math Standards
- How Long is the LSAT?

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject