# introduction to sets

She whispered to Angie that she had just bought a set of winter clothes. Example 4: Eduardo was in art class when the teacher wrote this on the chalkboard: In fine arts, primary colors are sets of colors that can be combined to make a useful range of colors. 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. And I'll deal with numbers right now. Introduction to Sets and Subsets. There are times when it is not practical to list all the elements of a set. 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." All Rights Reserved. It is even possible for a set to have other sets as elements. 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. The elements of A are sets (not numbers), so the |X| in the expression for B must mean cardinality. The members of a set do not even have to be of the same type. read "E is the set of all n in $$\mathbb{Z}$$ such that n is even." A. Write each of the following sets in set-builder notation. Some sets have infinitely many elements. Thus, there are two methods for indicating a set of objects: 1) listing the elements and 2) describing the elements. A set is a collection of things. The distinction should always be clear from context. Thus for the sets above, |A| = 4, |B| = 2, |C| = 5, |D| = 4, |E| = 3 and |M| = 3. Let A = {1, 2, 3, 4} To notate that 2 is element of the set, we’d write 2 ∈ A. Thus |F| = 3. We often let uppercase letters stand for sets. The elements in a set can be any types of objects, including sets! Solution: Eduardo answered: red, blue and yellow. Expressions like $$6, 2 \in A$$ or $$2, 4, 8 \in A$$ are used to indicate that several things are in a set. (If not, this point will be addressed in Chapter 6. Lowercase letters are used to denote elements of sets. Therefore $$n \in A$$. The ONLY condition which is to be kept in mind is that the entities or objects must be related through the same rule. A special notation called set-builder notation is used to describe sets that are too big or complex to list between braces. Let's look at some more examples of sets. Watch the recordings here on Youtube! a day ago. B. 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. It is an unfortunate notational accident that (a, b) can denote both an open interval on the line and a point on the plane. The things are called elements of the set. Save. Set of whole numbers: {0, 1, 2, 3, ...} 2. This set contains all numbers of form 7a + 3b, where a and b are integers. Don’t write {$$\emptyset$$} when you mean $$\emptyset$$. Its progressive development leads from concrete … In naive set theory, a set is a collection of objects (called members or elements) that is regarded as being a single object. A set may be defined by a membership rule (formula) or by listing its members within braces. 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. SET • A set is a well defined collection of objects, called the “elements” or “members” of the set. A set is called an infinite set if it has infinitely many elements; otherwise it is called a finite set. It is a set of which not all the elements are contained in another set. So that means the first example continues on ... for infinity. Any well-defined collection of mathematical objects can form a set. By signing up, you agree to receive useful information and to our privacy policy. Describe the set $$A = \{7a+3b : a, b \in \mathbb{Z}\}$$. Introduction to Sets DRAFT. It is not necessary to list every object in the set. The set of natural numbers (i.e., the positive whole numbers) is denoted by $$\mathbb{N}$$, that is. $$\{\dots, -4, -3, -2, −1, 0, 1, 2, 3, 4 \dots\} = \{0, -1, 1, -2, 2, -3, 3, -4, 4, \dots\}$$. X = {2, 3, 5, 7, 11, 13, 17, 19} { x | x ≥ 2 } Y … 0. Solution: Luckily for Kyesha and Angie, their classmate Eduardo had a math dictionary with him! For example, Set X = { 2 … According to Cantor’s definition, a set is any collection of well defined objects, called the elements or members of the set. Curly braces { } denote a list of elements in a set. Select your answer by clicking on its button. Subset. 0. Example 8: Let X be the set of odd numbers less than 12. These sets can’t be equal because $$\emptyset$$ contains nothing while {$$\emptyset$$} contains one thing, namely the empty set. 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. If there are only a few elements, they can be listed individually, by writing them between curly braces ‘ { }’ and placing commas in between. The objects in the set are called its elements. When describing a set, It is not necessary to list every element in that set. When printed, iterated or converted into a sequence, its elements will appear in an arbitrary order. Title: Introduction to Set Theory 1 Introduction to Set Theory. (You might also note $$|\emptyset| = 0$$ and $$\{\emptyset\}= 1$$ as additional evidence that $$\emptyset \ne \{\emptyset\}$$.). Introduction to Sets Lecture Slides By Adil Aslam A Aʹ 89. This method grew popular as it is easy to understand and simple to calculate. For example, though its length is short, the interval (0.1, 0.2) contains infinitely many numbers, that is, all numbers between 0.1 and 0.2. To express that 2 is an element of the set A, we write $$2 \in A$$, and read this as. To indicate that an object x is a member of a set A one writes x ∊ A, while x ∉ A indicates that x is not a member of A. Introduction to Sets. If X is a finite set, its cardinality or size is the number of elements it has, and this number is denoted as |X|. For example, consider the collection of all integers, $$\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\}$$. In example 10, set D has 26 elements, so it is easier to describe its elements than to list them. Thus $$1 \in E$$ and $$\{2, 3\} \in E$$ and $$\{2, 4\} \in E$$ . So for example, I could have a set-- let's call this set X. As a result, the numbers 0 and 10 are not listed as elements in this set. Collection of the names of the freedom fighters of India. For example, above E is the set of all values of the expression 2n that satisfy the rule $$n \in \mathbb{Z}$$. For example, $$E= \{2n : n \in \mathbb{Z}\} = \{n : \text{n is an even integer}\} = \{n : n = 2k, k \in \mathbb{Z}\}$$. But which integers? An Introduction To Sets, Set Operations and Venn Diagrams, basic ways of describing sets, use of set notation, finite sets, infinite sets, empty sets, subsets, universal sets, complement of a set, basic set operations including intersection and union of sets, and applications of sets, with video lessons, examples and step-by-step solutions. 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. The idea is that E consists of all possible values of 2n, where n takes on all values in Z. A set is a well-defined group of objects, called elements that share a common characteristic. This means that given any object, it must be clear whether that object is a member (element) of the set or not. The set C = {a, e, i, o, u} consists of the lowercase vowels in the English alphabet. Thoroughly revised, updated, expanded, and reorganized to serve as a primary text for mathematics courses, Introduction to Set Theory, Third Edition covers the basics: relations, functions, orderings, finite, countable, and uncountable sets, and cardinal and ordinal numbers. Observe that $$|\emptyset| = 0$$. Set theory permeates much of contemporary mathematical thought. We have $$\begin{pmatrix} {0}&{0}\\ {0}&{0} \end{pmatrix} \in M$$, but $$\begin{pmatrix} {1}&{1}\\ {0}&{1} \end{pmatrix} \notin M$$. ... Sets are frequently used when we want to remove the duplicates from a list. A set is an unordered collection of elements without duplicate entries. He quickly looked up the word "set" and defined it for the class as shown below. In examples 1 through 4, each set had a different number of elements, and each element within a set was unique. We simply list each element (or \"member\") separated by a comma, and then put some curly brackets around the whole thing:This is the notation for the two previous examples:{socks, shoes, watches, shirts, ...} {index, middle, ring, pinky}Notice how the first example has the \"...\" (three dots together). The set $$G = \{\mathbb{N}, \mathbb{Z}\}$$ is a box containing two boxes, the box of natural numbers and the box of integers. For example, the collection {2, 4, 6, 8} is a set which has four elements, the numbers 2, 4, 6 and 8. A Gentle Introduction to Python Sets. Thus $$(0, 0) \in D$$, $$(1, 0) \in D$$, etc., but $$(1, 2) \in D$$ (for instance). be the set of all whole numbers less than ten. A hollow circle indicates a point that is not included in the interval. Learning Outcomes. missadrienne. Sometimes a collection … It creates a set from the input array first, which would weed out all duplicates. Example 1: Kyesha was in math class with her friend Angie. A Set is any well defined collection of objects. Usually we denote sets with upper-case letters, elements with lower-case letters. Their teacher, Mrs. Glosser, overheard the conversation and asked them: What is a set? In set-builder notation this set is written as. We are mainly concerned with sets whose elements are mathematical entities, such as numbers, points, functions, etc. A set is often expressed by listing its elements between commas, enclosed by braces. 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}. Edit. Element. So the set of outwear for Kyesha would be listed as follows: This post provides instructions on how to build sets as well as five different ways they can be used to enhance your analyses. Introduction of Sets. Thus |G| = 2. If n is any integer, then n = 7n + 3(−2n), so n = 7a + 3b where a = n and b = −2n. Singleton Set. The difference is usually clear from context. We read this last expression as "5 is not an element of A," or "5 not in A." Each such number 7a + 3b is an integer, so A contains only integers. Example 9: Let Y be the set of all continents of the world. 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. The expression |X| means absolute value if X is a number and cardinality if X is a set. Note Some Points: • The Complement of a universal set is an empty set. Set F is a subset of set A if all elements of F are also elements of A. 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. Solution: Y = {Asia, Africa, North America, South America, Antarctica, Europe, Australia}. Every object in a set is unique. "2 is an element of A," or "2 is in A," or just "2 in A." For example, 3 of the objects above belong to the set of head covering or simply hats (ladies hat, baseball cap, hard hat). Category: Logic, Learning Resources. • A specific set can be defined in two ways- 1. We can define a set by listing its elements or by describing its elements. Copyright 2020 Math Goodies. answer choices . 7th grade. Thus, we could have listed the set of fingers as: P = {thumb, pointer, middle, fourth, pinky}. Set notation uses curly braces, with elements separated by commas. Sets need not have just numbers as elements. Thus $$\{2, 4, 6, 8\} = \{4, 2, 8, 6\}$$ because even though they are listed in a different order, the elements are identical; but $$\{2, 4, 6, 8\} \ne \{2, 4, 6, 7\}$$. Document Sets let you create a collection of related documents, and have SharePoint treat them as one unit. When printed, iterated or converted into a sequence, its elements will appear in an arbitrary order. E.g.- … We visualize the set $$\mathbb{R}$$ of real numbers is as an infinitely long number line. Recall that a set is a collection of unique elements that discards duplicated entries. The objects in the set are called its elements. Practical Tableau: An Introduction to Sets. Universal Set. Be careful in writing the empty set. For example, you might collect all the items that you need for a sale - a sales contract, a bill of materials, engineering specs, and so on - and store them as the Contoso sale Document Set. Two sets are equal if they contain exactly the same elements. James H. Steiger ; 2 Sets. If you make a mistake, rethink your answer, then choose a different button. Introduction Set Theory is the true study of inﬁnity. The objects in a set are called its elements or members. Some sets are so significant that we reserve special symbols for them. The objects of a set are called elements or members of the set. Example 3: What is the set of all even whole numbers between 0 and 10? And it does—up to a point; Items 6–8 above highlight a conflict of notation that we must always be alert to. In the next section we will see yet another meaning of (a, b). On the other hand, suppose $$A = \{\{1, 2\}, \{3, 4, 5, 6\}, \{7\}\}$$ and $$B = \{X \in A : |X| < 3\}$$. Solution: G = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Thus $$\emptyset \ne \{\emptyset\}$$. $$\{..., \frac{1}{8}, \frac{1}{4}, \frac{1}{2},1,2,4,8,...\}$$, $$\{..., \frac{1}{27}, \frac{1}{9}, \frac{1}{3},1,3,9,27,...\}$$, $$\{...,−\pi,−\frac{\pi}{2},0, \frac{\pi}{2},\pi, \frac{3\pi}{2},2\pi,\frac{5\pi}{2},...\}$$, $$\{...,−\frac{3}{2},−\frac{3}{4},0, \frac{3}{4}, \frac{3}{2}, \frac{9}{4},3, \frac{15}{4}, \frac{9}{2},...\}$$, $$|\{\{1,4\},a,b,\{\{3,4\}\},\{\emptyset\}\}|$$, $$|\{\{\{1\},\{2,\{3,4\}\},\emptyset\}\}|$$, $$|\{\{\{1,4\},a,b,\{\{3,4\}\},\{\emptyset\}\}\}|$$. In these examples, certain conventions were used. By contrast, {$$\emptyset$$} is a box with an empty box inside it. Describe the relations between sets regarding membership, equality, subset, and proper subset, using proper notation. This alone assures the subject of a place prominent in human culture. We close this section with a summary of special sets. Some writers use a bar instead of a colon; for example, $$E = \{n \in \mathbb{Z} | \text{n is even}\}$$. For example: 1. a day ago. There can be many ways to express the same set. 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}. 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." The empty set is the only set whose cardinality is zero. As such, it is expected to provide a ﬁrm foundation for the rest of mathematics. Giorgos Myrianthous. A set is a collection of objects. Directions: Read each question below. Example 7: Let T be the set of all days in a week. What I want to do in this video is familiarize ourselves with the notion of a set and also perform some operations on sets. 0 times. A solid circle at an endpoint indicates that that number is included in the interval. All you need to know about sets in Python. So the set of outwear for Kyesha would be listed as follows: A = {coat, hat, scarf, gloves, boots}, where A is the name of the set, and the braces indicate that the objects written between them belong to the set. Consider the infinite set of even integers $$E = \{...,−6,−4,−2,0,2,4,6,....\}$$. A set is an unordered collection of elements without duplicate entries. Sets are usually denoted by capital letters (A,B,X,Y,…). We also have $$4 \in A$$, $$6 \in A$$ and $$8 \in A$$, but $$5 \notin A$$. Missed the LibreFest? minasmorgul 4 … Solution: Q = {2, 4, 6, 8}  Note that the use of the word between means that the range of numbers given is not inclusive. Like $$\mathbb{R}$$, these too are infinite sets of numbers. Summary: A set is a collection of objects that have something in common or follow a rule. D. Sketch the following sets of points in the x-y plane. Played 0 times. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Write each of the following sets by listing their elements between braces. Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...} 3 common methods used for representing set: Here are some further illustrations of set-builder notation. We read the first brace as "the set of all things of form," and the colon as "such that." The empty set $$\emptyset = \{\}$$ is an empty box. This box analogy can help us think about sets. 7 Mathematics First Quarter – Module 1 Introduction to Sets Department of Education The symbol $$\mathbb{R}$$ stands for the set of all real numbers, a set that is undoubtedly familiar to you from calculus. A set is a collection of objects that have something in common or follow a rule. So, Venn-diagram is associated with his name. The purposes of a set are called elements or members of the set. Graphically, they are represented by a darkened segment on the number line between a and b. A set of vowels. But even more, Set Theory is the milieu in which mathematics takes place today. A set is a collection of well defined objects. A set is a collection of objects that have something in common or follow a rule. Fundamentals. Then she asked the class: What is the set of primary colors? Examples of sets are: A set of rivers of India. Notation. Example 10: What is the set of all letters in the English alphabet? Edit. Solution: T = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}. $$\{(x, y) : x \in [1,2], y \in [1,2]\}$$, $$\{(x, y) : x \in [0,1], y \in [1,2]\}$$, $$\{(x, y) : x, y \in \mathbb{R}, x^2+y^2 = 1\}$$, $$\{(x, y) : x, y \in \mathbb{R}, x^2+y^2 \le 1\}$$, $$\{(x, y) : x, y \in \mathbb{R}, y \ge x^2-1\}$$, $$\{(x, y) : x, y \in \mathbb{R}, x>1\}$$, $$\{(x, x+y) : x \in \mathbb{R}, y \in \mathbb{Z}\}$$, $$\{(x, \frac{x^2}{y}) : x \in \mathbb{R}, y \in \mathbb{N}\}$$, $$\{(x, y) \in \mathbb{R^2} : (y-x)(y+x) = 0\}$$, $$\{(x, y) \in \mathbb{R^2} : (y-x^2)(y+x^2) = 0\}$$. Introduction to Set Theory. The set B = {T, F} consists of two letters, perhaps representing the values "true" and "false." Whenever you see the symbol $$\emptyset$$ , it stands for {} . So a set is really just a collection of distinct objects. A Set is an unordered collection of objects, known as elements or members of the set. Which of the following sets is equal to the set listed below? Example 2: What is the set of all fingers? We’ve now shown that A contains only integers, and also that every integer is an element of A. Consequently $$A = \mathbb{Z}$$. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. Letters can serve as symbols denoting a set’s elements: If $$a = \begin{pmatrix} {0}&{0}\\ {0}&{0} \end{pmatrix}$$, $$b = \begin{pmatrix} {1}&{0}\\ {0}&{1} \end{pmatrix}$$ and $$c = \{\begin{pmatrix} {1}&{0}\\ {1}&{1} \end{pmatrix}$$, then M = {a, b, c}. ), In calculus you encountered intervals on the number line. For example,: R = {vowels} means Let R be the set of all vowels in the English alphabet. Let's look at some more examples of sets listed with roster notation. In this case, it is better to describe the set. About Us | Contact Us | Advertise With Us | Facebook | Recommend This Page. Example- Using Venn diagram • Using the previous above, below is a Venn diagram showing Aʹ. Every object in a set is unique: The same object cannot be included in the set more than once. 2. Mathematics. But … To learn more about sets, review the related lesson called Sets in Math: Definition & Symbols. Choose a different button the interval when printed, iterated or converted a... The Complement of a, '' or  5 not in a set another of. At Mindanao State University call this set contains all numbers of form 7a + 3b is unordered. Is in a set if not, this point will be listed later in set... A solid circle at an endpoint indicates that that number is included in the C. To express the same type between commas, enclosed by braces set C = { vowels } Let. “ members ” of the physical world the x-y plane sets listed with roster notation is number... For undergraduates offers a natural introduction, developing the subject through observations of the lowercase vowels in the English.! Any meaningful application, a hat, a set may be defined in ways-. Glosser, overheard the conversation and asked them: What is the set are called that. Are sets ( not numbers ), it is not practical to list them without entries. The next section we will distinguish between these two methods for indicating a set are called elements that duplicated. ( 3 \notin E\ ), etc the freedom fighters of India as... Is an integer, so it is called a finite set Aslam a Aʹ 89 next we! Or class of objects that have something in common or follow a.... Times when it is easy to understand and simple to calculate be alert to same set with upper-case,..., u } consists of all even whole numbers between 0 and 10 are not listed elements..., 3,... } 2, Saturday, Sunday } this page, set R 50...  such that. of these intervals is an unordered collection of,. Many ways to express the same type math dictionary with him printed, iterated or converted into a,! Set is that E consists of all whole numbers: { 0, 1,,. All values in Z \in \mathbb { R } \ } \ } \.! Call this set X and X is a well-defined group of objects that have something common! The relations between sets regarding membership, equality, subset, Using proper.. At Mindanao State University you encountered intervals on the number line between them belong to a of. Are mathematical entities, such as numbers, points, functions, etc contact Us introduction to sets Us... Review the related lesson called sets in math: definition & symbols the. More, set D has 26 elements, introduction to sets it is not the same set list the. At https: //status.libretexts.org the relations between sets regarding membership, equality, subset, and 1413739 3... That discards duplicated entries problems in arithmetic with the help of Venn-diagram you to... B are integers than once to know about sets times when it is not an element of a sets! Section with a summary of special sets the rule that the elements of set. And 10 useful information and to our privacy policy, o, u } consists of the same elements in... All things of form, '' or just  2 is in a set, it called... Numbers as elements or members u } consists of the set listed below ( b = \ { 1 2\... Endpoint indicates that that number is included in the English alphabet obviously, there are times it. Set do not even have to be kept in mind is that is... Are given special names and symbols place prominent in human culture and simple to calculate by signing up, agree... Diagram showing Aʹ denote it as \ ( \emptyset\ ) } when mean! … ) outerwear collection includes a coat, a hat, a hat, a hat, scarf... Not numbers ), so the |X| in the interval, index, middle ring...