{"id":663761,"date":"2023-07-04T12:05:41","date_gmt":"2023-07-04T06:35:41","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=663761"},"modified":"2025-05-15T15:23:55","modified_gmt":"2025-05-15T09:53:55","slug":"sets","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/topics\/sets\/","title":{"rendered":"Sets"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Introduction\" title=\"Introduction\">Introduction<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Definition_of_Sets_in_Mathematics\" title=\"Definition of Sets in Mathematics\">Definition of Sets in Mathematics<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Some_standard_mathematical_examples_of_Sets\" title=\"Some standard mathematical examples of Sets\">Some standard mathematical examples of Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Example_of_the_finite_Sets\" title=\"Example of the finite Sets\">Example of the finite Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Elements_of_a_set\" title=\"Elements of a set\">Elements of a set<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Cardinal_Number_of_a_Set\" title=\"Cardinal Number of a Set\">Cardinal Number of a Set<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Representation_of_Sets_in_Set_Theory\" title=\"Representation of Sets in Set Theory\">Representation of Sets in Set Theory<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Semantic_Form\" title=\"Semantic Form\">Semantic Form<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Roster_Form\" title=\"Roster Form\">Roster Form<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Set_Builder_Form\" title=\"Set Builder Form\">Set Builder Form<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Venn_Diagram_Method_of_Visual_Representation_of_Sets\" title=\"Venn Diagram: Method of Visual Representation of Sets\">Venn Diagram: Method of Visual Representation of Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Sets_Symbols\" title=\"Sets Symbols\">Sets Symbols<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Types_of_Sets\" title=\"Types of Sets\">Types of Sets<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Singleton_Sets\" title=\"Singleton Sets\">Singleton Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Finite_Sets\" title=\"Finite Sets\">Finite Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Infinite_Sets\" title=\"Infinite Sets\">Infinite Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Empty_or_Null_Sets\" title=\"Empty or Null Sets\">Empty or Null Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-18\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Equal_Sets\" title=\"Equal Sets\">Equal Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-19\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Unequal_Sets\" title=\"Unequal Sets\">Unequal Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-20\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Equivalent_Sets\" title=\"Equivalent Sets\">Equivalent Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-21\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Overlapping_Sets\" title=\"Overlapping Sets\">Overlapping Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-22\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Disjoint_Sets\" title=\"Disjoint Sets\">Disjoint Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-23\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Subset_and_Superset\" title=\"Subset and Superset\">Subset and Superset<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-24\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Universal_Set\" title=\"Universal Set\">Universal Set<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-25\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Power_Sets\" title=\"Power Sets\">Power Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-26\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Operations_on_Sets\" title=\"Operations on Sets\">Operations on Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-27\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Properties_of_Sets\" title=\"Properties of Sets\">Properties of Sets<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-28\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Sets_Solved_Examples\" title=\"Sets: Solved Examples\">Sets: Solved Examples<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-29\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#Frequently_Asked_Question_on_Sets\" title=\"Frequently Asked Question on Sets\">Frequently Asked Question on Sets<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-30\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#How_are_sets_defined_in_mathematics\" title=\"How are sets defined in mathematics?\">How are sets defined in mathematics?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-31\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#What_are_some_examples_of_sets\" title=\"What are some examples of sets?\">What are some examples of sets?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-32\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#What_are_the_elements_of_a_set\" title=\"What are the elements of a set?\">What are the elements of a set?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-33\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#What_is_the_roster_form_of_representing_sets\" title=\"What is the roster form of representing sets?\">What is the roster form of representing sets?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-34\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/sets\/#What_is_the_set_builder_form_of_representing_sets\" title=\"What is the set builder form of representing sets?\">What is the set builder form of representing sets?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Introduction\"><\/span>Introduction<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Mathematicians defined a Set as a collection of distinct objects forming a group. The set can have any group of items, a collection of numbers, a group of different alphabets, days of the week, types of vehicles, and many more. Each item in the set is called an Element of the set. Curly {} brackets are used while representing the set.<\/p>\n<p><strong>A few examples of the set are:<\/strong><\/p>\n<ul>\n<li><strong>Set A = {1,2,3,4,5}<\/strong><\/li>\n<li><strong>Set B = {a,e,i,o,u}<\/strong><\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Definition_of_Sets_in_Mathematics\"><\/span><strong>Definition of Sets in Mathematics<\/strong><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Mathematicians have defined a Set as a collection of distinct objects. Sets are named using capital letters. Also, in the set theory, a set&#8217;s elements can be the people&#8217;s names, letters of the alphabet, numbers, shapes, places, etc.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Some_standard_mathematical_examples_of_Sets\"><\/span>Some standard mathematical examples of Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Some <strong>standard examples of sets<\/strong> regularly used in mathematics are mentioned below.<\/p>\n<ul>\n<li>Set of natural numbers <strong>N= {1,2,3,4,5\u2026}<\/strong><\/li>\n<li>Set of whole numbers <strong>W= {0,1,2,3,4,5\u2026}<\/strong><\/li>\n<li>Set of integers <strong>Z = {&#8230;-3,-2,-1,0,1,2,3\u2026}<\/strong><\/li>\n<li>Set of the rationals <strong>Q = {p\/q where q is an integer and q is not equal to 0}<\/strong><\/li>\n<li>Set of the irrationals <strong>Q&#8217;= {x where x is not a rational number}<\/strong><\/li>\n<li>Set of real numbers <strong>R = QUQ&#8217;<\/strong><\/li>\n<\/ul>\n<p>The above-mentioned sets are infinite, but there can also be finite sets.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Example_of_the_finite_Sets\"><\/span>Example of the finite Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li>Set of even natural numbers less than 10: set A = {2,4,6,8}<\/li>\n<li>Set of vowels: set B = {a,e,i,o,u}<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Elements_of_a_set\"><\/span>Elements of a set<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>The items present in a set are called the elements of the set.<\/strong> The items in the set are closed in curly brackets. Commas separate the elements. To denote that an element is contained in a set, the symbol \u2208 is used; if not, then the \u2209 symbol is used.<\/p>\n<p>For example:<\/p>\n<p>Given set: Set A = {2,4,6,8}<\/p>\n<p>Therefore, 2 \u2208 Set A<\/p>\n<p>Also, 17 \u2209 Set A<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Cardinal_Number_of_a_Set\"><\/span>Cardinal Number of a Set<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A set&#8217;s cardinal number or order denotes the total number of elements in the set.<\/p>\n<p>For even natural numbers less than 10, n(A) = 4.<\/p>\n<p>Sets are a collection of unique elements, but each element must be related to another by a unique relation.<\/p>\n<p>For example, if we define a set for &#8220;the days of the week,&#8221; then each element will represent either of the seven days of the week; it cannot represent any month of the year.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Representation_of_Sets_in_Set_Theory\"><\/span>Representation of Sets in Set Theory<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Different <strong>set notation<\/strong>s are used to represent sets in set theory. The major difference between the three set notations is the way in which the elements are listed.<\/p>\n<p>The three different forms of representation of sets in mathematics are listed below:<\/p>\n<ul>\n<li>Semantic form<\/li>\n<li>Roster form<\/li>\n<li>Set builder form<\/li>\n<\/ul>\n<p>For a better understanding of the sets, refer to the table below.<\/p>\n<div class=\"table-responsive\">\n<table class=\"table table-bordered table-striped\" style=\"width: 100%; height: 120px;\" cellspacing=\"0\" cellpadding=\"5\">\n<tbody>\n<tr style=\"background-color: #89cff0; color: black;\">\n<td style=\"text-align: center; height: 48px;\" colspan=\"2\"><strong>Representation of set A defined as a set of the first five even natural numbers<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px;\">Semantic Form<\/td>\n<td style=\"height: 24px;\">A set of the first five even natural numbers<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px;\">Roster Form<\/td>\n<td style=\"height: 24px;\">{2,4,6,8,10}<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px;\">Set Builder Form<\/td>\n<td style=\"height: 24px;\">{x\u2208 N | x \u2264 10 and x is even}<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3><span class=\"ez-toc-section\" id=\"Semantic_Form\"><\/span>Semantic Form<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Semantic notation<\/strong> describes the set of elements using a statement.<br \/>\nFor example, set {1,2,3,4,5} is <strong>&#8220;the set of first five natural numbers&#8221;<\/strong> in the semantic notation.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Roster_Form\"><\/span>Roster Form<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The<strong> roster notation<\/strong> is the most common form to represent sets. Roster form is the form of representation in which the elements of the sets are enclosed in the {} and are separated by commas.<br \/>\nFor example, Set A = {1,2,3,4,5},<br \/>\nwhich is the collection of the first five natural numbers.<\/p>\n<p><strong>Points to note:<\/strong><\/p>\n<ul>\n<li>In a roster form, the set is independent of the order of the set.<\/li>\n<li>For example, the set of the first five natural numbers can also be defined as {2,5,1,4,3}.<\/li>\n<li>If there is an endless list of elements in a set, then a series of dots is used at the end of the last element to define the rest of the series.<\/li>\n<li>For example, infinite sets are represented as X = {1, 2, 3, 4, 5, 6, 7,&#8230;}, where X is the set of natural numbers going to infinity.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Set_Builder_Form\"><\/span>Set Builder Form<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The<strong> set builder notation<\/strong> uses a certain rule or a statement that specifies the common feature of a set&#8217;s elements.<br \/>\nFor example,<br \/>\nA = { k | k is an even number, k \u2264 10}<\/p>\n<ul>\n<li>The exact definition in roster Form is defined as A= {2,4,6,8,10}<\/li>\n<li>The statement says all the elements of set A are even numbers less than or equal to 20. Sometimes a &#8220;:&#8221; is used instead of the &#8220;|&#8221;.<\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Venn_Diagram_Method_of_Visual_Representation_of_Sets\"><\/span>Venn Diagram: Method of Visual Representation of Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A pictorial representation of sets is called <strong>Venn Diagram<\/strong>. Each set is represented in a pictorial representation, for instance, circles. The elements of a set are represented inside the used representation shape. Sometimes a rectangle which represents the universal set, encloses the circles. The Venn diagram also describes the relation between elements of various sets.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Sets_Symbols\"><\/span>Sets Symbols<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Set symbols are used to define the elements of a given set. The following table shows the set theory symbols and their meaning.<\/p>\n<div class=\"table-responsive\">\n<table class=\"table table-bordered table-striped\" cellspacing=\"0\" cellpadding=\"5\">\n<tbody>\n<tr style=\"background-color: #89cff0; color: black;\">\n<td style=\"text-align: center;\"><strong>Symbol<\/strong><\/td>\n<td style=\"text-align: center;\"><strong>Meaning<\/strong><\/td>\n<\/tr>\n<tr>\n<td>{}<\/td>\n<td>Symbol of set<\/td>\n<\/tr>\n<tr>\n<td>U<\/td>\n<td>Universal set<\/td>\n<\/tr>\n<tr>\n<td>n(X)<\/td>\n<td>Cardinal number of set X<\/td>\n<\/tr>\n<tr>\n<td>b \u2208 A<\/td>\n<td>&#8216;b&#8217; is an element of set A<\/td>\n<\/tr>\n<tr>\n<td>a \u2209 B<\/td>\n<td>&#8216;a&#8217; is not an element of set B<\/td>\n<\/tr>\n<tr>\n<td>\u2205<\/td>\n<td>Null or empty set<\/td>\n<\/tr>\n<tr>\n<td>A U B<\/td>\n<td>Set A union set B<\/td>\n<\/tr>\n<tr>\n<td>A \u2229 B<\/td>\n<td>Set A intersection set B<\/td>\n<\/tr>\n<tr>\n<td>A \u2286 B<\/td>\n<td>Set A is a subset of Set B<\/td>\n<\/tr>\n<tr>\n<td>B \u2287 A<\/td>\n<td>Set B is the superset of Set A<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/topics\/maths-topics\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 80px;\" type=\"button\">Maths Topics <\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/topics\/types-of-matrices\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 80px;\" type=\"button\">Types of Matrices: Examples, Class 12th, Identity Matrices, and More<\/button><\/a><\/p>\n<\/div>\n<h2><span class=\"ez-toc-section\" id=\"Types_of_Sets\"><\/span>Types of Sets<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>There are different types of sets in mathematical set theory. Each set represents a different type of collection of elements. A few basic types of sets are explained below.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Singleton_Sets\"><\/span>Singleton Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A set that consists of only one element is referred to as a singleton or unit set.<br \/>\nFor example, Set A = { k | k is an integer between 3 and 5}, which is A = {4}.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Finite_Sets\"><\/span>Finite Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A finite set is a set that has a limited or countable number of elements.<br \/>\nExample, Set B = {k | k is a prime number less than 10}, which is B = {2,3,5,7}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Infinite_Sets\"><\/span>Infinite Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>An infinite set is a set that contains an unlimited number of elements.<br \/>\nExample: Set A = {Multiples of 5}<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Empty_or_Null_Sets\"><\/span>Empty or Null Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>An empty set, also known as a null set, is a set that does not contain any elements. It is denoted by the symbol &#8216;\u2205&#8217;, read as &#8216;phi&#8217;.<br \/>\nExample: Set X = { } = \u2205<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Equal_Sets\"><\/span>Equal Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>When two sets have exactly the same elements, they are called equal sets.<br \/>\nExample: A = {1,8,4} and B = {1,4,8}. Here, set A and set B are equal sets.<br \/>\nThis can be represented as A = B.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Unequal_Sets\"><\/span>Unequal Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>If two sets have at least one different element, they are considered unequal sets.<br \/>\n<strong>Example<\/strong>: A = {1,8,4} and B = {2,5,4}.<br \/>\nHere, set A and set B are unequal sets. This can be represented as A \u2260 B.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Equivalent_Sets\"><\/span>Equivalent Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Equivalent sets refer to sets with the same number of elements, even though the elements may differ.<br \/>\n<strong>Example:<\/strong> A = {1,2,3} and B = {a,b,c}. Here, set A and set B are equivalent sets since n(A) = n(B)<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Overlapping_Sets\"><\/span>Overlapping Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Two sets overlap if there is at least one element common to both sets.<br \/>\n<strong>Example:<\/strong> A = {2,4,9} B = {4,7,10}.<br \/>\nHere, element 4 is present in set A and set B. Therefore, A and B are overlapping sets.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Disjoint_Sets\"><\/span>Disjoint Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Disjoint sets are sets that do not share any common elements. Example: A = {1,2,3,14} B = {5,6,7,11}. Here, set A and set B are disjoint sets.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Subset_and_Superset\"><\/span>Subset and Superset<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>When every element of set A is also present in set B, set A is considered a subset of set B (A \u2286 B), and set B is considered the superset of set A (B \u2287 A).<br \/>\n<strong>Example:<\/strong> Consider the sets A = {1,2,3} and B = {1,2,3,4,8,9,12}.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Universal_Set\"><\/span>Universal Set<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A universal set is the collection of all elements related to a particular subject. It is denoted by the letter &#8216;U&#8217;.<br \/>\n<strong>Example:<\/strong> Let U = {The list of all road transport vehicles}.<br \/>\nHere, a set of cars, cycles, and trains are all subsets of this universal set.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Power_Sets\"><\/span>Power Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The power set of a set is the collection of all possible subsets that the set can contain.<br \/>\n<strong>Example:<\/strong> Set A = {1,2,8}. Power set of A is = {\u2205, {1}, {2}, {8}, {1,2}, {2,8}, {1,8}, {1,2,8}}.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Operations_on_Sets\"><\/span>Operations on Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Some <strong>important operations on sets in set theory<\/strong> include union, intersection, difference, the complement of a set, and the cartesian product of a set. A brief explanation of set operations is as follows.<\/p>\n<div class=\"table-responsive\">\n<table class=\"table table-bordered table-striped\" cellspacing=\"0\" cellpadding=\"5\">\n<tbody>\n<tr style=\"background-color: #89cff0; color: black;\">\n<td style=\"text-align: center; height: 48px;\" colspan=\"3\"><strong>OPERATIONS ON SETS<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Union of Sets<\/td>\n<td>The union of sets is denoted as A U B. It lists the elements in both sets A and B.<\/td>\n<td>{1, 3} \u222a {1, 4} = {1, 3, 4}<\/td>\n<\/tr>\n<tr>\n<td>Intersection of Sets<\/td>\n<td>The intersection of sets, denoted by A \u2229 B, lists the elements common to both set A and set B.<\/td>\n<td>{1, 2} \u2229 {2, 4} = {2}<\/td>\n<\/tr>\n<tr>\n<td>Set Difference<\/td>\n<td>The set difference, denoted by A &#8211; B, lists the elements in set A that are absent in set B.<\/td>\n<td>A = {2, 3, 4} and B = {4, 5, 6}. A &#8211; B = {2, 3}<\/td>\n<\/tr>\n<tr>\n<td>Set Complement<\/td>\n<td>Set complement, denoted by A&#8217;, is the set of all elements in the universal set that are not present in set A. In other words, A&#8217; is denoted as U &#8211; A, which is the difference in the elements of the universal set and set A<\/td>\n<td>U= {1,2,3,4,5,6,7,8,9}<br \/>\nA= {1,3,5,7,9}<br \/>\nA\u2019= {2,4,6,8} = U &#8211; A<\/td>\n<\/tr>\n<tr>\n<td>Cartesian Product of Sets<\/td>\n<td>The cartesian product of two sets, denoted by A \u00d7 B, is the product of two non-empty sets, wherein ordered pairs of elements are obtained.<\/td>\n<td>{1, 3} \u00d7 {1, 3} = {(1, 1), (1, 3), (3, 1), (3, 3)}<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>Sets Formulas in Set Theory<br \/>\nSets find their application in algebra, statistics, and probability. There are some important set theory formulas in set theory. Refer to the table below for the Set formulas.<\/p>\n<div class=\"table-responsive\">\n<table class=\"table table-bordered table-striped\" style=\"width: 66.3826%; height: 264px;\" cellspacing=\"0\" cellpadding=\"5\">\n<tbody>\n<tr style=\"background-color: #89cff0; color: black;\">\n<td style=\"height: 24px;\"><strong>For any two overlapping sets, A and B<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; height: 24px;\">n(A U B) = n(A) + n(B) &#8211; n(A \u2229 B)<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"height: 24px; text-align: center;\">n (A \u2229 B) = n(A) + n(B) &#8211; n(A U B)<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; height: 24px;\">n(A) = n(A U B) + n(A \u2229 B) &#8211; n(B)<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; height: 24px;\">n(B) = n(A U B) + n(A \u2229 B) &#8211; n(A)<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; height: 24px;\">n(A &#8211; B) = n(A U B) &#8211; n(B)<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; height: 24px;\">n(A &#8211; B) = n(A) &#8211; n(A \u2229 B)<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"background-color: #89cff0; color: black; text-align: center;\"><strong>For any two sets A and B that are disjoint<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; height: 24px;\">n(A U B) = n(A) + n(B)<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; height: 24px;\">A \u2229 B = \u2205<\/td>\n<\/tr>\n<tr style=\"height: 24px;\">\n<td style=\"text-align: center; height: 24px;\">n(A &#8211; B) = n(A)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3><span class=\"ez-toc-section\" id=\"Properties_of_Sets\"><\/span>Properties of Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Similar to numbers, sets also have associative property, commutative property, etc. There are six important properties of sets.<br \/>\nGiven three sets, A, B, and C, the properties for these sets are as follows.<\/p>\n<table class=\"table table-bordered table-striped\" cellspacing=\"0\" cellpadding=\"5\">\n<tbody>\n<tr style=\"background-color: #89cff0; color: black;\">\n<td><strong>Property of Set<\/strong><\/td>\n<td><strong>Example<\/strong><\/td>\n<\/tr>\n<tr>\n<td>Commutative Property<\/td>\n<td>A U B = B U A<br \/>\nA \u2229 B = B \u2229 A<\/td>\n<\/tr>\n<tr>\n<td>Associative Property<\/td>\n<td>(A \u2229 B) \u2229 C = A \u2229 (B \u2229 C)<br \/>\n(A U B) U C = A U (B U C)<\/td>\n<\/tr>\n<tr>\n<td>Distributive Property<\/td>\n<td>A U (B \u2229 C) = (A U B) \u2229 (A U C)<br \/>\nA \u2229 (B U C) = (A \u2229 B) U (A \u2229 C)<\/td>\n<\/tr>\n<tr>\n<td>Identity Property<\/td>\n<td>A U \u2205 = A<br \/>\nA \u2229 U = A<\/td>\n<\/tr>\n<tr>\n<td>Complement Property<\/td>\n<td>A U A&#8217; = U<\/td>\n<\/tr>\n<tr>\n<td>Idempotent Property<\/td>\n<td>A \u2229 A = A<br \/>\nA U A = A<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3><span class=\"ez-toc-section\" id=\"Sets_Solved_Examples\"><\/span>Sets: Solved Examples<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><strong>Example 1.<\/strong> Let A = {1, 2, 7} and B = {3, 4, 5, 6}. Find the union of sets A and B.<br \/>\nSolution: The union of sets A and B, denoted by A \u222a B, is the set that contains all the elements that are in A or B (or both).<br \/>\nIn this case, A \u222a B = {1, 2, 3, 4, 5, 6, 7}.<\/p>\n<p><strong>Example 2. <\/strong>Let A = {1, 2, 4} and B = {3, 4, 5, 6}. Find the intersection of sets A and B.<br \/>\nSolution: The intersection of sets A and B, denoted by A \u2229 B, is the set that contains all the elements that are common to both A and B.<br \/>\nIn this case, A \u2229 B = {4}.<\/p>\n<p><strong>Example 3. <\/strong>Let A = {1, 2, 4} and B = {3, 4, 5, 6}. Find the relative complement of set B with respect to set A.<br \/>\nSolution: The relative complement of set B with respect to set A, denoted by A &#8211; B, is the set containing all the elements in A but not in B.<br \/>\nIn this case, A &#8211; B = {1, 2}.<\/p>\n<p><strong>Example 4.<\/strong> Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Determine whether A is a subset of B.<br \/>\nSolution: A is a subset of B if all the elements of A are also elements of B.<br \/>\nIn this case, A is not a subset of B since A contains elements (1, 2) absent in B.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Frequently_Asked_Question_on_Sets\"><\/span>Frequently Asked Question on Sets<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"How_are_sets_defined_in_mathematics\"><\/span>How are sets defined in mathematics?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tSets are defined as a well-defined collection of objects. The objects can be people's names, letters of the alphabet, numbers, shapes, places, etc. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_are_some_examples_of_sets\"><\/span>What are some examples of sets?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tExamples of sets include the set of natural numbers (N), the set of whole numbers (W), the set of integers (Z), the set of rational numbers (Q), and the set of real numbers (R). \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_are_the_elements_of_a_set\"><\/span>What are the elements of a set?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe elements of a set are the individual objects or items that belong to the set. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_roster_form_of_representing_sets\"><\/span>What is the roster form of representing sets?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tRoster form represents sets by listing the elements of the set inside curly brackets, separated by commas. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_the_set_builder_form_of_representing_sets\"><\/span>What is the set builder form of representing sets?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tSet builder form uses a rule or statement to specify the common feature of the set's elements. It uses a vertical bar and a condition to describe the elements.\t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"How are sets defined in mathematics?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Sets are defined as a well-defined collection of objects. The objects can be people's names, letters of the alphabet, numbers, shapes, places, etc.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What are some examples of sets?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Examples of sets include the set of natural numbers (N), the set of whole numbers (W), the set of integers (Z), the set of rational numbers (Q), and the set of real numbers (R).\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What are the elements of a set?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The elements of a set are the individual objects or items that belong to the set.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the roster form of representing sets?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Roster form represents sets by listing the elements of the set inside curly brackets, separated by commas.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is the set builder form of representing sets?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Set builder form uses a rule or statement to specify the common feature of the set's elements. It uses a vertical bar and a condition to describe the elements.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>Introduction Mathematicians defined a Set as a collection of distinct objects forming a group. The set can have any group [&hellip;]<\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"sets","_yoast_wpseo_title":"What is Sets in Math - Types of Sets, Operations, and Solved Examlpes","_yoast_wpseo_metadesc":"Sets - collection of distinct objects. Sets are named using capital letters. 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Sets are named using capital letters. 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