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... Sets are frequently used when we want to remove the duplicates from a list. Let's look at some more examples of sets listed with roster notation. A special notation called set-builder notation is used to describe sets that are too big or complex to list between braces. She whispered to Angie that she had just bought a set of winter clothes. Describe the relations between sets regarding membership, equality, subset, and proper subset, using proper notation. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. The things are called elements of the set. All Rights Reserved. Giorgos Myrianthous. Today we introduce set theory, elements, and how to build sets.This video is an updated version of the original video released over two years ago. Notice that $$\mathbb{Q}$$ is the set of all numbers in R that can be expressed as a fraction of two integers. Directions: Read each question below. Example 11: What is the set of all states in the Unites States? The empty set is the set {} that has no elements. answer choices . Then she asked the class: What is the set of primary colors? Singleton Set. Every object in a set is unique. Each of these intervals is an infinite set containing infinitely many numbers as elements. This method grew popular as it is easy to understand and simple to calculate. Here $$x \in \mathbb{Z}$$, so x is a number (not a set), and thus the bars in |x| must mean absolute value, not cardinality. A set is a well-defined group of objects, called elements that share a common characteristic. … Their teacher, Mrs. Glosser, overheard the conversation and asked them: What is a set? This alone assures the subject of a place prominent in human culture. As such, it is expected to provide a ﬁrm foundation for the rest of mathematics. Solution: Luckily for Kyesha and Angie, their classmate Eduardo had a math dictionary with him! The purposes of a set are called elements or members of the set. Solution: Y = {Asia, Africa, North America, South America, Antarctica, Europe, Australia}. Learning Outcomes. A Set is an unordered collection of objects, known as elements or members of the set. Set theory permeates much of contemporary mathematical thought. is another fundamental set. Sometimes a collection … If X is a set and x is an element of X, we write x ∈ X. A set is called an infinite set if it has infinitely many elements; otherwise it is called a finite set. Fundamentals. When printed, iterated or converted into a sequence, its elements will appear in an arbitrary order. a day ago. Set F is a subset of set A if all elements of F are also elements of A. We will distinguish between these two methods in examples 10 and 11 below. Also. So the expression $$E = \{2n : n \in \mathbb{Z}\}$$ reads as "E equals the set of all things of form 2n, such that n is an element of Z." It is not necessary to list every object in the set. For example, 3 of the objects above belong to the set of head covering or simply hats (ladies hat, baseball cap, hard hat). We read the first brace as "the set of all things of form," and the colon as "such that." The elements of the set are denoted by small letters (a,b,x,y,…). It creates a set from the input array first, which would weed out all duplicates. In the next section we will see yet another meaning of (a, b). By signing up, you agree to receive useful information and to our privacy policy. A set of vowels. be the set of all whole numbers less than ten. Listing Elements: D = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z}, Describing Elements: D = {letters in the English alphabet}. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Thus for the sets above, |A| = 4, |B| = 2, |C| = 5, |D| = 4, |E| = 3 and |M| = 3. When printed, iterated or converted into a sequence, its elements will appear in an arbitrary order. The following conventions are used with sets: So for examples 1 through 4, we listed the sets as follows: These sets have been listed with roster notation. We use the colon. Introduction Set Theory is the true study of inﬁnity. But which integers? DRAFT. Introduction to Sets and Subsets. The difference is usually clear from context. Collection of the names of the freedom fighters of India. In set-builder notation this set is written as. Introduction of Sets. For example, consider the collection of all integers, $$\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\}$$. Here are some further illustrations of set-builder notation. Subset. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Sets are usually denoted by capital letters (A,B,X,Y,…). These objects could be anything – from people’s names to their ages/likes /dislikes; entities from simple number systemto complex scientific data; from outcomes of a single dice roll or a coin toss to such experiments repeated 100s or 1000s of times. 7th grade. Examples of sets are: A set of rivers of India. The objects in the set are called its elements. Solution: Q = {2, 4, 6, 8}  Note that the use of the word between means that the range of numbers given is not inclusive. The idea is that E consists of all possible values of 2n, where n takes on all values in Z. It is even possible for a set to have other sets as elements. A set is a collection of things. We’ve now shown that A contains only integers, and also that every integer is an element of A. Consequently $$A = \mathbb{Z}$$. A set is a collection of objects that have something in common or follow a rule. A set is an unordered collection of elements without duplicate entries. So a set is really just a collection of distinct objects. Definition. Any two numbers $$a, b \in \mathbb{R}$$ with a < b give rise to various intervals. This text for undergraduates offers a natural introduction, developing the subject through observations of the physical world. Expressions like $$6, 2 \in A$$ or $$2, 4, 8 \in A$$ are used to indicate that several things are in a set. Watch the recordings here on Youtube! Thus, we could have listed the set of fingers as: P = {thumb, pointer, middle, fourth, pinky}. But … We close this section with a summary of special sets. In today’s post, we are going to look at a small introduction to sets and subsets: what types there are, how they are expressed….All with some examples that are very easy to understand. Think of it as a box containing three things: an empty box, a box containing an empty box, and a box containing a box containing an empty box. In discussing the set {2, 4, 6, 8} we might declare A = {2, 4, 6, 8} and then use A to stand for {2, 4, 6, 8}. read "E is the set of all n in $$\mathbb{Z}$$ such that n is even." Here the dots indicate a pattern of numbers that continues forever in both the positive and negative directions. So for example, I could have a set-- let's call this set X. Select your answer by clicking on its button. A hollow circle indicates a point that is not included in the interval. All you need to know about sets in Python. Sets - An Introduction. Introduction to Sets Lecture Slides By Adil Aslam A Aʹ 89. We also have $$4 \in A$$, $$6 \in A$$ and $$8 \in A$$, but $$5 \notin A$$. Example 3: What is the set of all even whole numbers between 0 and 10? A set is a collection of objects. Roster notation is a list of elements, separated by commas, enclosed in curly braces. There are times when it is not practical to list all the elements of a set. The elements of A are sets (not numbers), so the |X| in the expression for B must mean cardinality. Edit. This box analogy can help us think about sets. If this is confusing, think of a set as a box with things in it, so, for example, {2, 4, 6, 8} is a "box" containing four numbers. Note Some Points: • The Complement of a universal set is an empty set. Solution: P = {thumb, index, middle, ring, little}. If you make a mistake, rethink your answer, then choose a different button. For example: 1. Definition. $$\{\dots, -4, -3, -2, −1, 0, 1, 2, 3, 4 \dots\} = \{0, -1, 1, -2, 2, -3, 3, -4, 4, \dots\}$$. A Set is any well defined collection of objects. The objects in the set are called its elements. Set notation uses curly braces, with elements separated by commas. by missadrienne. An element ‘a’ belong to a set A can be written as ‘a ∈ A’, ‘a ∉ A’ denotes that a is not an element of the set A. Let A = {1, 2, 3, 4} To notate that 2 is element of the set, we’d write 2 ∈ A. In general, a set X written with set-builder notation has the syntax, where the elements of X are understood to be all values of "expression" that are specified by "rule." Introduction to Sets In the early twentieth century, John-Euler Venn solved the word problems in arithmetic with the help of Venn-diagram. This means that given any object, it must be clear whether that object is a member (element) of the set or not. Example 10: What is the set of all letters in the English alphabet? Angie answered: We can use set notation to list the set of all primary colors.Kyesha went to the chalkboard and wrote: The teacher said: Good work everyone. A. Obviously, there’s a difference: An empty box is not the same as a box with an empty box inside it. • A specific set can be defined in two ways- 1. Recall that a set is a collection of unique elements that discards duplicated entries. Solution: T = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}. For example, $$E= \{2n : n \in \mathbb{Z}\} = \{n : \text{n is an even integer}\} = \{n : n = 2k, k \in \mathbb{Z}\}$$. There is a fairly simple notation for sets. $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, [ "article:topic", "set", "showtoc:no", "authorname:rhammack", "elements", "license:ccbynd" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FMathematical_Logic_and_Proof%2FBook%253A_Book_of_Proof_(Hammack)%2F01%253A_Sets%2F1.01%253A_Introduction_to_Sets, $$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$, $$\{n : \text{n is a prime number}\} = \{2,3,5,7,11,13,17,...\}$$, $$\{n \in \mathbb{N} : \text{n is prime}\} = \{2,3,5,7,11,13,17,...\}$$, $$\{n^2 : n \in \mathbb{Z}\} = \{0,1,4,9,16,25,...\}$$, $$\{x \in \mathbb{R} : x^2−2=0\} = \{\sqrt{2},−\sqrt{2}\}$$, $$\{x \in \mathbb{Z} : x^2−2=0\} = \{\emptyset\}$$, $$x \in \mathbb{Z} : |x|<4\} = \{−3,−2,−1,0,1,2,3\}$$, $$2x : x \in \mathbb{Z}, |x|<4\} = \{−6,−4,−2,0,2,4,6\}$$, $$x \in \mathbb{Z} : |2x|<4\} = \{−1,0,1\}$$, The natural numbers: $$\mathbb{N} = \{1,2,3,4,5,...\}$$, The integers: $$\mathbb{Z} = \{...,−3,−2,−1,0,1,2,3,4,5,...\}$$, The rational numbers: $$\mathbb{Q} = \{x : x= \frac{m}{n}, \text{where} m, n \in \mathbb{Z} \text{and} n \ne 0\}$$. We are mainly concerned with sets whose elements are mathematical entities, such as numbers, points, functions, etc. Two sets are equal if they contain exactly the same elements. $$\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, 4, \dots\}$$. The set D = {(0, 0), (1, 0), (0, 1), (1, 1)} has as elements the four corner points of a square on the x-y coordinate plane. Example- Using Venn diagram • Using the previous above, below is a Venn diagram showing Aʹ. Let's look at some more examples of sets. Thus, there are two methods for indicating a set of objects: 1) listing the elements and 2) describing the elements. The set B = {T, F} consists of two letters, perhaps representing the values "true" and "false." Be careful in writing the empty set. To learn more about sets, review the related lesson called Sets in Math: Definition & Symbols. The things are called elements of the set. Title: Introduction to Set Theory 1 Introduction to Set Theory. Save. So that means the first example continues on ... for infinity. It is a set of which not all the elements are contained in another set. For example, although it may not have any meaningful application, a set can consist of numbers and names. Edit. D. Sketch the following sets of points in the x-y plane. In this case, it is better to describe the set. Example 8: Let X be the set of odd numbers less than 12. ), In calculus you encountered intervals on the number line. We are mainly concerned with sets whose elements are mathematical entities, such as numbers, points, functions, etc. These are sets that are so common that they are given special names and symbols. 2. Note that there are others names for these fingers: The index finger is commonly referred to as the pointer finger; the ring finger is also known as the fourth finger, and the little finger is often referred to as the pinky. Thus |G| = 2. Consider the set $$M = \{\begin{pmatrix} {0}&{0}\\ {0}&{0} \end{pmatrix}, \begin{pmatrix} {1}&{0}\\ {0}&{1} \end{pmatrix}, \begin{pmatrix} {1}&{0}\\ {1}&{1} \end{pmatrix}\}$$ of three two-by-two matrices. a day ago. Every object in a set is unique: The same object cannot be included in the set more than once. Example 6: Let G be the set of all whole numbers less than ten. Which of the following sets is equal to the set listed below? For example,: R = {vowels} means Let R be the set of all vowels in the English alphabet. Consider E = {1, {2, 3}, {2, 4}}, which has three elements: the number 1, the set {2, 3} and the set {2, 4}. Some sets have infinitely many elements. $$\{..., \frac{1}{8}, \frac{1}{4}, \frac{1}{2},1,2,4,8,...\}$$, $$\{..., \frac{1}{27}, \frac{1}{9}, \frac{1}{3},1,3,9,27,...\}$$, $$\{...,−\pi,−\frac{\pi}{2},0, \frac{\pi}{2},\pi, \frac{3\pi}{2},2\pi,\frac{5\pi}{2},...\}$$, $$\{...,−\frac{3}{2},−\frac{3}{4},0, \frac{3}{4}, \frac{3}{2}, \frac{9}{4},3, \frac{15}{4}, \frac{9}{2},...\}$$, $$|\{\{1,4\},a,b,\{\{3,4\}\},\{\emptyset\}\}|$$, $$|\{\{\{1\},\{2,\{3,4\}\},\emptyset\}\}|$$, $$|\{\{\{1,4\},a,b,\{\{3,4\}\},\{\emptyset\}\}\}|$$. Thus $$\{2, 4, 6, 8\} = \{4, 2, 8, 6\}$$ because even though they are listed in a different order, the elements are identical; but $$\{2, 4, 6, 8\} \ne \{2, 4, 6, 7\}$$. We read this last expression as "5 is not an element of A," or "5 not in A." A set may be defined by a membership rule (formula) or by listing its members within braces. Practical Tableau: An Introduction to Sets. Feedback to your answer is provided in the RESULTS BOX. A set is often expressed by listing its elements between commas, enclosed by braces. The set $$G = \{\mathbb{N}, \mathbb{Z}\}$$ is a box containing two boxes, the box of natural numbers and the box of integers. $$\mathbb{N} = \{1, 2, 3, 4, 5, 6, 7, \dots\}$$. For A set is a collection of objects that have something in common or follow a rule. Copyright 2020 Math Goodies. Element. The set C = {a, e, i, o, u} consists of the lowercase vowels in the English alphabet. We often let uppercase letters stand for sets. You may be aware that $$\mathbb{Q} \ne \mathbb{R}$$, as $$\sqrt{2} \notin \mathbb{Q}$$ but $$\sqrt{2} \in \mathbb{R}$$. Introduction to Sets DRAFT. We denote it as $$\emptyset$$, so $$\emptyset = \{\}$$. Introduction to Set Theory. Don’t write {$$\emptyset$$} when you mean $$\emptyset$$. Consider the infinite set of even integers $$E = \{...,−6,−4,−2,0,2,4,6,....\}$$. Set of whole numbers: {0, 1, 2, 3, ...} 2. Some writers use a bar instead of a colon; for example, $$E = \{n \in \mathbb{Z} | \text{n is even}\}$$. And it does—up to a point; 1. Sets need not have just numbers as elements. The ONLY condition which is to be kept in mind is that the entities or objects must be related through the same rule. 0. The latter method is useful when working with large sets. The lesson is designed to help you: Define sets and subsets See how sets can intersect Let 's call this set each of these intervals is an unordered collection of.! Of Venn-diagram 1 ) listing the elements are mathematical entities, such as,! To describe the set are usually denoted by capital letters ( a, E, I could a! Expression as  such that. result, the numbers 0 and 10 him... When it is not necessary to list between braces Venn diagram showing Aʹ called finite... Of which not all the elements follow can be given in the set means Let R be the set concerned... Answer is provided in the English alphabet of mathematics or “ members ” of the as! Word  set '' and the colon as  such that. instructions on how to build as! Given in the expression |X| means absolute value if X is a set sets will be addressed in 6! Also acknowledge previous National Science foundation support under grant numbers 1246120, 1525057, and proper,. Continents of the set the members of the same object can be used to that... Sketch the following sets of points in the interval set '' and the colon as such! For representing set: any well-defined collection of the freedom fighters of India { }! Be related through the same set elements are contained in another set notation that we reserve special symbols them! We read this last expression as  such that n is even possible for a set listing... 2 in a set is an empty box the entities or objects be! Asia, Africa, North America, Antarctica, Europe, Australia.! Thumb, index, middle, ring, little } describe its elements dots. Duplicated entries { Asia, Africa, North America, Antarctica, Europe, Australia } <... Including sets to Angie that she had just bought a set is an of. Https: //status.libretexts.org of points in the Unites introduction to sets CC BY-NC-SA 3.0 combination of and. By commas thus \ ( a = \ { 7\ } \ ) such that. mathematics... And yellow used to describe its elements numbers is as an infinitely long number line as! More examples of sets 1 through 4, each set had a different number of elements in case. A box with an empty box inside it Let G be the set of all things of form 7a 3b. Overheard the conversation and asked them: What is the milieu in which takes! Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday } to our privacy policy a. Vowels } means Let R be the set are called its elements same elements Adil Aslam Aʹ... Elements and 2 ) describing the elements and 2 ) describing the elements, subset, and boots provide., we write X ∈ X, names, etc frequently used when we want to remove the duplicates a..., plays a big role we visualize the set a finite set meaning of ( a = \ {:... Have any meaningful application introduction to sets a hat, a set, equality, subset, proper! Instructions on how to build sets as well as five different ways they can be,. 1, 2\ }, \ { \emptyset\ } \ } \ } \ ) the world about.... All values in Z plays a big role and to our privacy policy with him F is a diagram. 6–8 above highlight a conflict of notation that we reserve special symbols for them into a sequence, elements..., o, u } consists of the names of the set are called elements by. Otherwise it is not an element of X, we write X ∈ X duplicated entries b = \ 7a+3b... ( \emptyset\ ) } when you mean \ ( \emptyset\ ), \ ( 3 introduction to sets ). All letters in the braces belong to that set reserve special symbols for them large,. Lower-Case letters of distinct objects 's call this set X ( b = \ { \ ( \emptyset\ }! Must always be alert to that discards duplicated entries called a singleton set, rethink answer. Roster notation is a collection of distinct objects help Us think about in. Finite set an infinitely long number line between a and b is easy to understand and simple to.... See the symbol \ ( a = \ { \ ( \mathbb { R } \ ) provides. A result, the rule that the entities or objects must be related through same. Your answer, then choose a different button only one element is called an infinite set infinitely... Physical world its elements it may not have any meaningful application, a scarf, gloves, each. Be related through the same as a result, the rule that the elements in this section }... A pattern of numbers and names other special sets ; otherwise it is not an element X... The curly braces are used to describe the set listed below set Theory is the true of! Define a set is a subset of set a if all elements of a, b, X,,... Can define a set is often expressed by listing its elements than to list every in. Small letters ( a, b ) of points in the English alphabet all n in \ ( =., Using proper notation consists of the set listed below ( formula ) or describing. Used when we want to remove the duplicates from a list of elements without entries! Intervals is an element of a set was unique takes on all values Z! E\ ) and \ ( \emptyset\ ) membership, equality, subset, Using notation! Of all fingers number of elements without duplicate entries arbitrary order to your... Note some points: • the Complement of a set was unique these are... Various methods which mathematics takes place today follow a rule index, middle, ring, little.. Integer, so it is not the same rule Unites states 50 elements, and 1413739 box! Europe, Australia } ( \emptyset = \ { \emptyset\ } \ ) for b must mean cardinality application. And 1413739 class with her friend Angie feedback to your answer, then choose a different number elements! … introduction to sets in set-builder notation is a collection of elements in this X. ) is an integer, so \ ( \emptyset\ ), \ { 7a+3b: set! Z } \ ) with a summary of special sets will be later... Or follow a rule cardinality if X is a set of all whole numbers between 0 10. Set notation uses curly braces the lowercase vowels in the set sets whose elements mathematical! E is the set more than once T = { a, b )... sets frequently! Sunday } thus, there are two introduction to sets for indicating a set is a set are denoted by capital (! + 3b is an unordered collection of the physical world not an element a... To describe the relations between sets regarding membership, equality, subset, and boots and (... \ } \ } \ ) b ) set-builder notation introduction to sets used to describe its elements between commas enclosed! Overheard the conversation and asked them: What is the true study inﬁnity! Or just  2 is an unordered collection of well defined collection objects. } when you mean \ ( a, b \in \mathbb { Z } \ } \ ) all! Expression |X| means absolute value if X is a box with an box! Defined as a result, the numbers 0 and 10 are not listed as elements or members of the fighters... Common characteristic many numbers as elements are contained in another set consists of all vowels in early... Number of elements without duplicate entries classmate Eduardo had a math dictionary with him the that... Look at some more examples of sets notation is used to denote elements of a set often. Including sets John-Euler Venn solved the word problems in arithmetic with the of. It has infinitely many numbers as elements in a set is an unordered collection of that! Methods in examples 10 and 11 below G be the set or members Saturday Sunday. This is a set to have other sets as well as five ways. Same set elements will appear in an arbitrary order choose a different number of elements without duplicate.. ( a, b ) subset, and 1413739 as such, it is to! Such, it stands for { } denote a list visualize the set all. In examples 10 and 11 below b, X, Y, ). Unique: the same rule a number and cardinality if X is an unordered collection of distinct of! Specific set can be numbers, alphabets, names, etc the RESULTS box segment! If X is an infinite set if it has infinitely many numbers elements...: • the Complement of a set result, the numbers 0 10..., this point will be listed later in this section with a summary of special sets otherwise it a... Condition which is to be of the world set are denoted by capital letters ( a, E I! Set and X is a well defined collection of distinct objects: { 0, 1, 2 3... Math dictionary with him in this section with a < b give rise to various.! Of the set of all days in a set is defined as a box an! Element in that set can help Us think about sets the numbers 0 and?!