{"id":141520,"date":"2022-09-13T14:46:14","date_gmt":"2022-09-13T19:46:14","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=141520"},"modified":"2025-12-26T15:34:22","modified_gmt":"2025-12-26T21:34:22","slug":"intro-to-set-theory","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/intro-to-set-theory\/","title":{"rendered":"Introduction to Set Theory"},"content":{"rendered":"<h1>Introduction to Set Theory<\/h1>\n\n\t\t\t<div id=\"mmDeferVideoEncompass_ODOOPDh9-f8\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_ODOOPDh9-f8\" data-source-videoID=\"ODOOPDh9-f8\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Introduction to Set Theory Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Introduction to Set Theory\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_ODOOPDh9-f8:hover {cursor:pointer;} img#videoThumbnailImage_ODOOPDh9-f8 {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1779-thumb-final-3-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_ODOOPDh9-f8\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_ODOOPDh9-f8\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Introduction to Set Theory\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_ODOOPDh9-f8\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_ODOOPDh9-f8\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_ODOOPDh9-f8\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>In algebra and calculus, you\u2019ll sometimes come across sets of numbers, which are usually denoted with special symbols like curly brackets and the \u201celement\u201d symbol.<\/p>\n<div class=\"examplesentence\">{\u2009 } , \\(\\in \\) \u2009\n<\/div>\n<p>\n&nbsp;<br \/>\nEven though at first glance these unfamiliar symbols can be intimidating, you\u2019ll soon see that set theory is a helpful tool that is not too difficult to master.<\/p>\n<p>When we talk about sets in mathematics, we are just talking about collections of objects. For example, I could have a set called \u201c<em>A<\/em>\u201d which contains the even numbers from 2 to 8.<\/p>\n<div class=\"examplesentence\">\\(A =\\) {\\(2, 4, 6, 8\\)}<\/div>\n<p>\n&nbsp;<\/p>\n<p>Or I could have a set \u201c<em>B<\/em>\u201d containing all integer multiples of 5.<\/p>\n<div class=\"examplesentence\">\\(B=\\){\\(\u2026, -10, -5, 0, 5, 10, \u2026\\)}<\/div>\n<p>\n&nbsp;<\/p>\n<p>We can create sets for any collection of objects or numbers. So if you were interested in writing down the ages of everyone in your class, you could express those numbers together as a set. In both of the example sets <em>A<\/em> and <em>B<\/em>, notice that we started by writing the name of the set, followed by an equals sign, and then wrote the elements of the set, separated by commas, between a pair of curly brackets. The numbers in a set are referred to as the <strong>elements<\/strong> of that set, and we use the element sign \\((\\in )\\) to show that a number belongs to a set. For example, we can say that \u201c2 is an element of <em>A<\/em>,\u201d because 2 is one of the numbers in set <em>A<\/em>. Likewise, we can say that \u201c5 is an element of <em>B<\/em>.\u201d <\/p>\n<div class=\"examplesentence\">\\(2\\in  A\\)           &nbsp;                 \\(5\\in  B\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>Sometimes it is helpful to note that a particular number does <em>not<\/em> belong to a set. To denote this, we just write the element symbol with a strikethrough. For example, the number 2 is not in set <em>B<\/em>, so we would write \\(2\\not \\in  B\\). Likewise, the number 5 is not in set <em>A<\/em>, so we can write \\(5\\not \\in A\\).<\/p>\n<p>One of the interesting properties of sets in mathematics is that all sets include something called \u201cthe <strong>empty set<\/strong>.\u201d The empty set is exactly what it sounds like; it has no elements and iis denoted using this symbol: \\(\u2205\\). You know how whenever you have \\(x\\) in an equation, it\u2019s understood to mean you have one \\(x\\)? The 1 is understood, and therefore doesn\u2019t need to be written. In the same way, it is understood that all sets contain the empty set. One way to think about this is to say \u201cthe set <em>A<\/em> has 2, 4, 6, 8, and nothing else.\u201d The \u201cnothing else\u201d is the empty set, so \\(\u2205\\in  A\\). And of course, \\(\u2205\\in B\\) as well.<\/p>\n<p>We\u2019ve already talked about using the element symbol to show that one number is contained in a set. If we wanted to show that multiple items belonged to a set, we could use the <strong>subset<\/strong> symbol, which looks like this: \\(\\subseteq \\). For example, we can say that the numbers 2, 4, and 6 are all in A by writing \\({2, 4, 6} \\subseteq A\\). Additionally, let\u2019s say we had a set <em>C<\/em> which contained only positive multiples of 5.<\/p>\n<div class=\"examplesentence\">\\(C =\\){\\(5, 10, 15, \u2026\\)}<\/div>\n<p>\n&nbsp;<\/p>\n<p>We could write that <em>C<\/em> is a subset of <em>B<\/em> because everything in <em>C<\/em> is included in <em>B<\/em>.<\/p>\n<div class=\"examplesentence\">\\(C \\subseteq B\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>Because we said the empty set is understood to be in every set, we can write that the empty set is a subset of every set too.<\/p>\n<div class=\"examplesentence\">\\(\u2205\\subseteq A\\)<br \/>\n&nbsp;<br \/>\n\\(\u2205\\subseteq B\\)<br \/>\n&nbsp;<br \/>\n\\(\u2205\\subseteq C\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>Let\u2019s say we have a set called \\(A*\\) which is identical to <em>A<\/em>.<\/p>\n<div class=\"examplesentence\">\\(A*={2,4,6,8}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>Because everything in \\(A*\\) is included in <em>A<\/em>, we can say that \\(A*\\subseteq A\\). Likewise, everything in <em>A<\/em> is included in \\(A*\\), so \\(A\\subseteq A*\\). <\/p>\n<p>When a subset is not equal to the entire set, we say that it is a <strong>proper subset<\/strong> and we can use this symbol: \\(\\subset \\). This is kind of like using a less-than or greater-than symbol instead of a less-than-or-equal-to, or greater-than-or-equal-to symbol. A subset symbol with an underline is saying \u201csubset or equal to.\u201d<\/p>\n<p>Earlier we wrote \\(C\\subseteq B\\), which is true. But we can also write \\(C\\subset B\\), since the set <em>C<\/em> does not contain all of the elements of <em>B<\/em>.<\/p>\n<p>A similar term you may come across is <strong>uperset<\/strong>, which is basically the opposite of a subset. We said earlier that the set \\({2, 4, 6}\\) is a subset of <em>A<\/em>. Another way of saying this is that <em>A<\/em> is a superset of \\({2,4,6}\\). We write this with a mirrored subset symbol: \\(A \\supseteq {2,4,6}\\).<\/p>\n<p>Let\u2019s go through a couple quick examples of what we\u2019ve learned so far. For the set \\(D=\\){\\(1, 2, 3, 4, 5\\)}, which of the following statements is true?<\/p>\n<div class=\"examplesentence\">\\(4\\in   D\\)     &nbsp;          \\(4\\notin  D\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>The true statement is the one on the left: 4 is an element of <em>D<\/em>, because 4 is included in the set <em>D<\/em>.<\/p>\n<p>Let\u2019s say we have a set \\(E =\\){\\(1, 2, 3, 4, 5, 6\\)}. Which of the following statements is true about <em>E<\/em>?<\/p>\n<div class=\"examplesentence\">\\(E\\subset D\\)        &nbsp;       \\(E\\subseteq  D\\)      &nbsp;         \\(E\\nsubseteq D\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>Even though <em>D<\/em> and <em>E<\/em> are similar, <em>E<\/em> includes the number 6, unlike <em>D<\/em>. Because not all elements of <em>E<\/em> are in <em>D<\/em>,the last option is correct. <em>E<\/em> is not a subset of <em>D<\/em>.<\/p>\n<p>Sometimes it can be helpful to visualize sets and their overlaps using Venn diagrams. Most commonly, Venn diagrams look like two overlapping circles, but they can feature three or more circles if necessary, and some circles may lie fully within others!<\/p>\n<p>Let\u2019s consider a regular two-circle Venn diagram, where the left circle is a set called <em>L<\/em> and the right circle is a set called <em>R<\/em>.<\/p>\n<p>If we wanted to label the small area of overlap between <em>L<\/em> and <em>R<\/em>, we would call it \u201cthe <strong>intersection<\/strong> of <em>L<\/em> and <em>R<\/em>,\u201d which is written with an upside-down U-shape: <em>L<\/em>\\(\\cap\\)  <em>R<\/em>. Intersections represent only the elements which are common to both sets. This is kind of like how we call the area where two roads cross each other an intersection; that area belongs to both roads!<\/p>\n<p>If, on the other hand, we wanted to talk about everything in <em>L<\/em> together with everything in <em>R<\/em>, we would say \u201cthe union of <em>L<\/em> and <em>R<\/em>,\u201d which is written with a U-like symbol: <em>L<\/em> \\(\\cup\\) <em>R<\/em>. Unions are used to <em>unite<\/em> the elements from two sets.<\/p>\n<p>Like some operations in mathematics, union and intersection follow the commutative, associative, and distributive properties. Let\u2019s quickly review what each of those mean. The <strong>commutative property<\/strong> tells us that \\(A\\cap B=B\\cap A\\), and similarly that \\(A\\cup B=B\\cup A\\). <\/p>\n<div class=\"examplesentence\"><span style=\"text-decoration: underline;\">Commutative<\/span><br \/>\n&nbsp;<br \/>\n\\(A\\cap B = B\\cap A\\)<br \/>\n&nbsp;<br \/>\n\\(A\\cup B = B\\cup A\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>The <strong>associative property<\/strong> tells us that \\((A\\cap B)\\cap C=A\\cap (B\\cap C)\\); likewise, \\((A\\cup B)\\cup C=A\\cup (B\\cup C)\\). <\/p>\n<div class=\"examplesentence\"><span style=\"text-decoration: underline;\">Associative<\/span><br \/>\n&nbsp;<br \/>\n\\((A\\cap B)\\cap C=A\\cap (B\\cap C)\\)<br \/>\n&nbsp;<br \/>\n\\((A\\cup B)\\cup C=A\\cup (B\\cup C)\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>Finally, the <strong>distributive property<\/strong> lets us rewrite \\(A\\cup (B\\cap C)\\) as \\((A\\cup B)\\cap (A\\cup C)\\). Similarly, \\(A\\cap (B\\cup C)=(A\\cap B)(A\\cap C)\\). <\/p>\n<div class=\"examplesentence\"><span style=\"text-decoration: underline;\">Distributive<\/span><br \/>\n&nbsp;<br \/>\n\\(A\\cup (B\\cap C)=(A\\cup B)\\cap (A\\cup C)\\)<br \/>\n&nbsp;<br \/>\n\\(A\\cap (B\\cap C)=(A\\cap B)\\cup (A\\cap C)\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>Feel free to pause the video now and take a moment to write each of these properties down.<\/p>\n<p>Another helpful term to know with set theory is called the <strong>universal set<\/strong>, which is often called <em>U<\/em>. It contains all elements of sets in consideration without repeats, and usually it comes into play when Venn diagrams are used. For example, this Venn diagram has sets <em>G<\/em> and <em>H<\/em>, as well as some elements which are neither in <em>G<\/em> nor <em>H<\/em>. Here, <em>U<\/em> contains all of the elements featured in the diagram: \\(U=\\){\\(-3,-2,-1,0,2,4,5,6,7\\)}.<\/p>\n<p>Sometimes it can be helpful to refer to all elements except those in a particular set. This can be done using the <strong>complement<\/strong> of a set. There are different ways to denote the complement of a set; some people use an apostrophe, while others may use a horizontal bar over the set, and still others use the letter <em>C<\/em>. In this diagram, we could show the complement of <em>G<\/em> as the set of all elements not in <em>G<\/em>.<\/p>\n<div class=\"examplesentence\">\\(G&#8217;=\\overline{G}=G^{C}=\\){\\(2,4,5,6,7\\)}<\/div>\n<p>\n&nbsp;<\/p>\n<p>Similarly, the complement of <em>H<\/em> can be found by selecting all elements in the universal set which are not in <em>H<\/em>.<\/p>\n<div class=\"examplesentence\">\\(H&#8217;=\\overline{H}=H^{C}=\\){\\(-1,-3,5,7\\)}<\/div>\n<p>\n&nbsp;<\/p>\n<p>Similar to the idea of complements, sometimes you need to find elements which belong to one set but not to another. To do this, we use a technique called \u201c<strong>set subtraction<\/strong>,\u201d which is denoted using a backslash. For example, <em>G<\/em> minus <em>H<\/em> can be written like this (<em>G<\/em>\u2216<em>H<\/em>) and includes the elements that are in <em>G<\/em> but not in <em>H<\/em>. This would ignore the elements \\(0\\) and \\(&#8211;\\)2, leaving us with only \\(-1\\) and \\(-3\\).<\/p>\n<div class=\"examplesentence\">\\(G \u2216H = \\){\\(-1, -3\\)}<\/div>\n<p>\n&nbsp;<\/p>\n<p>Similarly, <em>H<\/em> minus <em>G<\/em> would include all elements of <em>H<\/em> but those that are also in <em>G<\/em>.<\/p>\n<div class=\"examplesentence\">\\(H \u2216G = \\){\\(2, 4, 6\\)}<\/div>\n<p>\n&nbsp;<\/p>\n<p>Let\u2019s wrap up by naming some common sets that you\u2019ll likely come across in your assignments. We know that the <strong>natural numbers<\/strong>, which are also called the <strong>counting numbers<\/strong>, are whole numbers starting with 1. This set of numbers is abbreviated with a fancy letter N.<\/p>\n<div class=\"examplesentence\">\\(\\mathbb{N} = \\){\\(1, 2, 3, 4, \u2026\\)}<\/div>\n<p>\n&nbsp;<\/p>\n<p>The <strong>integers<\/strong> are whole numbers, including both positives and negatives, as well as zero. The set of integers is written as a fancy Z, which comes from the German word \u201cZahlen\u201d for numbers.<\/p>\n<div class=\"examplesentence\">\\(\\mathbb{Z} = \\){\\(\u2026, -2, -1, 0, 1, 2, \u2026\\)}<\/div>\n<p>\n&nbsp;<\/p>\n<p>The set of <strong>rational numbers<\/strong>, which are all numbers that can be expressed as fractions of integers and are not endless in their digits like pi, is called Q. The rational numbers are denoted by a Q because they can all be written as quotients.<\/p>\n<div class=\"examplesentence\">\\(\\mathbb{Q}=\\) all rational numbers<\/div>\n<p>\n&nbsp;<\/p>\n<p>The set of <strong>real numbers<\/strong> includes all rational and irrational numbers, and is denoted with the letter R.<\/p>\n<div class=\"examplesentence\">\\(\\mathbb{R}= \\)all real numbers<\/div>\n<p>\n&nbsp;<\/p>\n<p>Finally, the set of <strong>complex numbers<\/strong> includes all real and imaginary numbers, and is denoted with a C.<\/p>\n<div class=\"examplesentence\">\\(\\mathbb{C}=\\) all complex numbers<\/div>\n<p>\n&nbsp;<\/p>\n<p>Note that because all natural numbers are integers, all integers are rational numbers, all rational numbers are real, and the real numbers are a subset of complex numbers, we can write it this way:<\/p>\n<div class=\"examplesentence\">\\(NZQRC\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>Hearing all of this information now probably seems like a lot, so feel free to watch through this video a second time if necessary. As with all things though, this will come to you more naturally with practice. Now that you\u2019ve got a foundation for understanding set theory, you\u2019re ready to try some example problems on your own!<\/p>\n<p>I hope this video was helpful. Thanks for watching, and happy studying!<\/p>\n<\/div>\n<\/div>\n\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/discrete-math\/\"><strong>Return to Discrete Math Videos<\/strong><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Introduction to Set Theory Return to Discrete Math Videos<\/p>\n","protected":false},"author":22,"featured_media":141523,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-141520","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-discrete-math-videos","7":"page_category-video-pages-for-study-course-sidebar-ad","8":"page_type-video"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/141520","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=141520"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/141520\/revisions"}],"predecessor-version":[{"id":283762,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/141520\/revisions\/283762"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/141523"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=141520"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}