{"id":2087,"date":"2022-03-12T03:43:31","date_gmt":"2022-03-12T03:43:31","guid":{"rendered":"https:\/\/swatilathia.com\/?page_id=2087"},"modified":"2022-07-21T04:05:45","modified_gmt":"2022-07-21T04:05:45","slug":"set-theory","status":"publish","type":"page","link":"https:\/\/swatilathia.com\/?page_id=2087","title":{"rendered":"Set &#038; its types"},"content":{"rendered":"<body>\n<figure class=\"wp-block-image size-large is-resized is-style-default\"><img data-recalc-dims=\"1\" decoding=\"async\" src=\"https:\/\/i0.wp.com\/swatilathia.com\/wp-content\/uploads\/2022\/07\/image.png?resize=207%2C207&#038;ssl=1\" alt=\"\" class=\"wp-image-2515\" width=\"207\" height=\"207\" loading=\"lazy\" srcset=\"https:\/\/i0.wp.com\/swatilathia.com\/wp-content\/uploads\/2022\/07\/image.png?resize=1024%2C1024&amp;ssl=1 1024w, https:\/\/i0.wp.com\/swatilathia.com\/wp-content\/uploads\/2022\/07\/image.png?resize=300%2C300&amp;ssl=1 300w, https:\/\/i0.wp.com\/swatilathia.com\/wp-content\/uploads\/2022\/07\/image.png?resize=150%2C150&amp;ssl=1 150w, https:\/\/i0.wp.com\/swatilathia.com\/wp-content\/uploads\/2022\/07\/image.png?resize=768%2C768&amp;ssl=1 768w, https:\/\/i0.wp.com\/swatilathia.com\/wp-content\/uploads\/2022\/07\/image.png?w=1144&amp;ssl=1 1144w\" sizes=\"auto, (max-width: 207px) 100vw, 207px\" \/><\/figure>\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<label for=\"ez-toc-cssicon-toggle-item-69f685d90a11d\" class=\"ez-toc-cssicon-toggle-label\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><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><\/span><\/label><input type=\"checkbox\"  id=\"ez-toc-cssicon-toggle-item-69f685d90a11d\"  aria-label=\"Toggle\" \/><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#Set\" >Set<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#Method_of_Set_Representation\" >Method of Set Representation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#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-4\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#Finite_Set\" >Finite Set<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#Infinite_Set\" >Infinite 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:\/\/swatilathia.com\/?page_id=2087\/#Singleton_Set\" >Singleton 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:\/\/swatilathia.com\/?page_id=2087\/#Empty_Set\" >Empty Set<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#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-9\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#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-10\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#Subset\" >Subset<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#Proper_Subset\" >Proper Subset<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#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-13\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#Power_Set\" >Power Set<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/swatilathia.com\/?page_id=2087\/#Disjoint_Sets\" >Disjoint Sets<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Set\"><\/span>Set<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"has-medium-font-size wp-block-list\"><li>A set is a collection of well defined &amp; well distinguished objects. Simply, we can say, a set is a group of objects. <\/li><li>For example, a set of all the students study in BCA, a set of natural numbers \u2013 1,2,3,4.. , a set of odd numbers \u2013 1,3,5,7.. <\/li><li>All the objects of a set are also called elements of members of a set. <\/li><li>These elements are placed between curly braces \u2013 <strong>{ }<\/strong>. <\/li><li>The name of a set is always an Uppercase Letter. You can use lowercase letters as its elements such as <strong>A = { a, b, c, d }<\/strong>.<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"zak-oembed-container\"><div class=\"jetpack-video-wrapper\"><iframe loading=\"lazy\" title=\"Introduction of Set Theory\" width=\"812\" height=\"457\" src=\"https:\/\/www.youtube.com\/embed\/2BqpxS_wQRc?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/div><\/div>\n<\/div><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Method_of_Set_Representation\"><\/span>Method of Set Representation<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p class=\"has-medium-font-size\">There are mainly two types to represent a set :  (1) Roster method or Listing method (2) Rule method or Set Builder form or Property method<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Types_of_Sets\"><\/span>Types of Sets<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Finite_Set\"><\/span>Finite Set<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">A set which contains definite number of elements is called a finite set. For example, a set of month name in a year , {January, February, March, April, May, June, July, August, September, October, November, December}, a set of vowels in English alphabets {a, e, i, o, u}, a set of even numbers less than 10 {2, 4, 6, 8} are all finite set. Sometimes the number of elements of a set are long, still it is called finite set.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Infinite_Set\"><\/span>Infinite Set<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">A set which has no finite number of elements is called an infinite set. Let\u2019s say a set of all natural numbers {1,2,3,4,\u2026}, a set of positive integers which are divisible by 4 {4, 8, 12, 16,..}. In both examples, we can not say what the last element is.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Singleton_Set\"><\/span>Singleton Set<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">A set which has one &amp; only one element is called singleton set. For example : A={19}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Empty_Set\"><\/span>Empty Set<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">If a set does not have any element, then it is called empty set or null set. It denotes by <strong>{ }<\/strong> or <strong>\u03a6<\/strong> (phi). For example : B = { }<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"zak-oembed-container\"><div class=\"jetpack-video-wrapper\"><iframe loading=\"lazy\" title=\"Method of Representing Set | Roster and Listing Method | Types of set - Part 1\" width=\"812\" height=\"457\" src=\"https:\/\/www.youtube.com\/embed\/0hSJZfULJig?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/div><\/div>\n<\/div><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equal_Sets\"><\/span>Equal Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">Two sets A &amp; B are said to be equal set, if each element of set A is an element of set B, &amp; each element of set B is an element of set A. We can write A = B.<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Symbolically, if <strong>(\u2200x) (x \u2208 A =&gt; x \u2208 B) &amp; (\u2200x) (x \u2208 B =&gt; x \u2208 A)<\/strong>, Then<strong> A = B<\/strong><\/p>\n\n\n\n<p class=\"has-medium-font-size\">We can also say, if <strong>A \u2286 B &amp; B \u2286 A<\/strong> then <strong>A = B<\/strong><\/p>\n\n\n\n<p class=\"has-medium-font-size\">Let A = {10, 20, 30} &amp; B = {10, 20, 30}. Each element of A is in B and each element of B is in A. So A = B<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Another example, if A = {10, 20, 30} &amp; B = {10, 20, 30, 40}. Here each element of A is in B, but one element of B (40) is not there in A. So A \u2260 B.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Equivalent_Sets\"><\/span>Equivalent Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">If the number of elements in set A &amp; the number of elements in set B are same, then we can say both sets are equivalent. You can also say if it is possible to match each element of set A with one element of set B, then both sets are said to be equivalent sets. It is denoted by \u2245<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Let\u2019s say, A = {1, 2, 3} &amp; B  = {x, y, z}. Here both the sets have same number of elements, that is 3. So A \u2245 B<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Subset\"><\/span>Subset<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">A set A is said to be a subset of set B, if each element of set A is an element of set B. Set B is called the super set. <\/p>\n\n\n\n<p class=\"has-medium-font-size\">Symbolically, <strong>(\u2200x) (x \u2208 A =&gt; x \u2208 B)<\/strong> , then <strong>A \u2282 B<\/strong>. In notation, <strong>A \u2282 B<\/strong> <strong>if and only if,<\/strong> <strong>x \u2208 A =&gt; x \u2208 B<\/strong>.<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Let\u2019s say, A = {a, b} &amp; B = {a, b, c}. Here every element of set A is included in set B. So, A \u2282 B<\/p>\n\n\n\n<ul class=\"has-medium-font-size wp-block-list\"><li>Remember that, every set is a subset of that set itself , that means for A, <strong>A \u2286 A<\/strong>. <\/li><li>Null set is a subset of every set, that means for A, <strong>\u03a6  \u2282 A<\/strong>. <\/li><li>There are total <strong>2<sup>n<\/sup><\/strong> , n \u2208 N subset of every set. <strong>If A \u2282 B &amp; B \u2282 C, then A \u2282 C<\/strong>.<\/li><li>For example, A = {a, b} =&gt; Number of elements = 2<sup>2<\/sup> = 4. So subset of Set A = { }, {a}, {b}, {a, b}<\/li><\/ul>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"zak-oembed-container\"><div class=\"jetpack-video-wrapper\"><iframe loading=\"lazy\" title=\"Types of Set | Equal Set | Equivalent Set | Subset with examples\" width=\"812\" height=\"457\" src=\"https:\/\/www.youtube.com\/embed\/fVOxw14fdb4?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/div><\/div>\n<\/div><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Proper_Subset\"><\/span>Proper Subset<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">If all the elements of set A are the elements of set B, but at least one element of super set B is not an element of set A, then set A is known as proper subset of super set B.<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Let\u2019s say, A = {a, b} &amp; B = {a, b, c}. Here every element of set A is included in set B. So, A \u2282 B<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Universal_Set\"><\/span>Universal Set<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">The universal set in any discussion is the totality of elements under consideration as elements of set. It is denoted by <strong>U<\/strong>.<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Let\u2019s say, <strong>U<\/strong> = {1,2,3,4,5,6,7,8,9,10,11,12} , A = {3,6,9} &amp; B = {4,8,12}. Here all the elements of A &amp; B are in set <strong>U<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Power_Set\"><\/span>Power Set<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">The family of all the subsets of a given set is called power set. It is denoted by P().<\/p>\n\n\n\n<p class=\"has-medium-font-size\">Say, A = { 10, 20, 30 } then P(A) = { {10, 20, 30}, {10}, {20}, {30}, {10, 20}, {10, 30}, {20, 30}, { } }<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<div class=\"zak-oembed-container\"><div class=\"jetpack-video-wrapper\"><iframe loading=\"lazy\" title=\"Types of Set | Universal Set | Proper Subset | Power Set with examples\" width=\"812\" height=\"457\" src=\"https:\/\/www.youtube.com\/embed\/PEAwnwDpGIg?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/div><\/div>\n<\/div><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Disjoint_Sets\"><\/span>Disjoint Sets<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p class=\"has-medium-font-size\">Any two sets are said to be disjoint sets if their intersection is null set. A &amp; B are said to disjoint if (A \u2229 B) = \u03a6<\/p>\n<\/body>","protected":false},"excerpt":{"rendered":"<p>Set A set is a collection of well defined &amp; well distinguished objects. Simply, we can say, a set is a group of objects. For example, a set of all the students study in BCA, a set of natural numbers \u2013 1,2,3,4.. , a set of odd numbers \u2013 1,3,5,7.. All the objects of a [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"om_disable_all_campaigns":false,"footnotes":""},"class_list":["post-2087","page","type-page","status-publish","hentry"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/swatilathia.com\/index.php?rest_route=\/wp\/v2\/pages\/2087","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/swatilathia.com\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/swatilathia.com\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/swatilathia.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/swatilathia.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2087"}],"version-history":[{"count":9,"href":"https:\/\/swatilathia.com\/index.php?rest_route=\/wp\/v2\/pages\/2087\/revisions"}],"predecessor-version":[{"id":2522,"href":"https:\/\/swatilathia.com\/index.php?rest_route=\/wp\/v2\/pages\/2087\/revisions\/2522"}],"wp:attachment":[{"href":"https:\/\/swatilathia.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2087"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}