{"id":86455,"date":"2021-07-21T09:59:52","date_gmt":"2021-07-21T14:59:52","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=86455"},"modified":"2026-03-26T11:46:57","modified_gmt":"2026-03-26T16:46:57","slug":"negation","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/negation\/","title":{"rendered":"Negation"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_f0YW67wnnZ4\" 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_f0YW67wnnZ4\" data-source-videoID=\"f0YW67wnnZ4\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Negation Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Negation\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_f0YW67wnnZ4:hover {cursor:pointer;} img#videoThumbnailImage_f0YW67wnnZ4 {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1793-negation-1-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_f0YW67wnnZ4\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_f0YW67wnnZ4\" 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\",\"Negation\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_f0YW67wnnZ4\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_f0YW67wnnZ4\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_f0YW67wnnZ4\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction IL5_Function() {\n  var x = document.getElementById(\"IL5\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"IL5_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"IL5\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_Negation\" class=\"smooth-scroll\">What is Negation?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#How_Negation_Works\" class=\"smooth-scroll\">How Negation Works<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Multiple_Negations\" class=\"smooth-scroll\">Multiple Negations<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Negation_Practice_Questions\" class=\"smooth-scroll\">Negation Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hi, and welcome to this video about negation!<\/p>\n<p>In this video, we will explore what negation means, how it works, and how to work with multiple negations.<\/p>\n<p>Let\u2019s get started!<\/p>\n<h2><span id=\"What_is_Negation\" class=\"m-toc-anchor\"><\/span>What is Negation?<\/h2>\n<p>\nFirst, let\u2019s review a bit.<\/p>\n<p>Negation is a part of propositional logic. Remember that the primary purpose of propositional logic is to determine whether declarative statements\u2014arguments\u2014are true or false.<\/p>\n<p>Here is a quick reminder of some of the symbols that we use:<\/p>\n<table class=\"ATable\" style=\"margin: auto; width: 80%;\">\n<thead>\n<tr>\n<th><strong>Symbol<\/strong><\/th>\n<th><strong>Meaning<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Lower case letter<\/td>\n<td>variable<\/td>\n<\/tr>\n<tr>\n<td>Upper case letter<\/td>\n<td>predicate<\/td>\n<\/tr>\n<tr>\n<td>\\(\u2200\\)<\/td>\n<td>universal quantifier &#8220;for every&#8221;<\/td>\n<\/tr>\n<tr>\n<td>\\(\u2203\\)<\/td>\n<td>&#8220;there exists&#8221;<\/td>\n<\/tr>\n<tr>\n<td>:<\/td>\n<td>&#8220;such that&#8221;<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nFor example, the statement <span style=\"font-style:normal; font-size:90%\">\\(\\exists x:x^{2}= 9\\)<\/span> is true, since 3 is an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value that makes the equation true. If <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span> is our variable and <span style=\"font-style:normal; font-size:90%\">\\(P(x)\\)<\/span> is the predicate <span style=\"font-style:normal; font-size:90%\">\\(x^{2}= 9\\)<\/span>, then our statement looks like: <span style=\"font-style:normal; font-size:90%\">\\(\\exists xP(x)\\)<\/span>.<\/p>\n<p>On the other hand <span style=\"font-style:normal; font-size:90%\">\\(\\forall x:x^{2}=9\\)<\/span>, read \u201cFor every <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value, <span style=\"font-style:normal; font-size:90%\">\\(x^{2}= 9\\)<\/span>.\u201d is clearly false, because there are plenty of <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-values that make the equation false. Fully symbolic, this statement reads: <span style=\"font-style:normal; font-size:90%\">\\(\\forall xP(x)\\)<\/span>.<\/p>\n<h32>What Does Negation Mean?<\/h3>\n<p>\nNow, what exactly does negation mean? Negation means basically what it sounds like \u2013 to make a statement negative. Any statement can be negated.<\/p>\n<p>The word not is perhaps most commonly used to negate a statement, but other words and phrases can be used as well, such as:<\/p>\n<ul>\n<li>No<\/li>\n<li>It is false<\/li>\n<li>It is not the case<\/li>\n<li>It is not true<\/li>\n<\/ul>\n<p>When a statement is negated, its truth value is the opposite of what it was. Sometimes a truth table can be a helpful illustration:<\/p>\n<table class=\"ATable\" style=\"margin: auto;\">\n<thead>\n<tr>\n<th><strong>\\(P(x)\\)<\/strong><\/th>\n<th><strong>\\(\u00acP(x)\\)<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>True<\/td>\n<td>False<\/td>\n<\/tr>\n<tr>\n<td>False<\/td>\n<td>True<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"How_Negation_Works\" class=\"m-toc-anchor\"><\/span>How Negation Works<\/h2>\n<p>\nLet\u2019s revisit the statement: \u201cThere exists an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value such that <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>,\u201d which we\u2019ve said is true. Negating it should produce a false statement, which it does: \u201cThere does not exist an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value such that <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>.\u201d<\/p>\n<p>Note that negation of this statement is not \u201cThere exists an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value such that <span style=\"font-style:normal; font-size:90%\">\\(x^{2}\\neq 9\\)<\/span>,\u201d because <span style=\"font-style:normal; font-size:90%\">\\(x^{2}\\neq 9\\)<\/span> is a different predicate for a different time. We are testing <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-values that make <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span> true or false, not  <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-values that make  <span style=\"font-style:normal; font-size:90%\">\\(x^{2}\\neq 9\\)<\/span> true or false.<\/p>\n<p>Likewise, the statement: \u201cFor every <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value, <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>,\u201d which we\u2019ve said is false, is made true by negation: \u201cIt is not true for every <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value that <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>\u201d<\/p>\n<p>The symbol for negation is \u00ac. Let\u2019s add it to our list:<\/p>\n<table class=\"ATable\" style=\"margin: auto; width: 80%;\">\n<thead>\n<tr>\n<th><strong>Symbol<\/strong><\/th>\n<th><strong>Meaning<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Lower case letter<\/td>\n<td>variable<\/td>\n<\/tr>\n<tr>\n<td>Upper case letter<\/td>\n<td>predicate<\/td>\n<\/tr>\n<tr>\n<td>\\(\u2200\\)<\/td>\n<td>universal quantifier &#8220;for every&#8221;<\/td>\n<\/tr>\n<tr>\n<td>\\(\u2203\\)<\/td>\n<td>&#8220;there exists&#8221;<\/td>\n<\/tr>\n<tr>\n<td>:<\/td>\n<td>&#8220;such that&#8221;<\/td>\n<\/tr>\n<tr>\n<td>\u00ac<\/td>\n<td>&#8220;not,&#8221; negation<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nNow, back to our examples. We negated the statement \u201cThere exists an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value such that <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>,\u201d by saying \u201cThere does not exist an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value such that <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>.\u201d <\/p>\n<p>Symbolically, we originally said <span style=\"font-style:normal; font-size:90%\">\\(\\exists xP(x)\\)<\/span>, which is true. It can be negated by saying<\/p>\n<ul>\n<li><span style=\"font-style:normal; font-size:90%\">\\(\u00ac\\exists xP(x)\\)<\/span> (There does not exist an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value such that <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>) or<\/li>\n<li><span style=\"font-style:normal; font-size:90%\">\\(\\forall x\u00acP(x)\\)<\/span>, (For all <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>, <span style=\"font-style:normal; font-size:90%\">\\(x^{2}\\)<\/span> never equals 9)<\/li>\n<\/ul>\n<p>Both the negations have the same meaning and are truth-equivalent (in this case, both false).<\/p>\n<p>In our second example, we began by saying \u201cFor every <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value, <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>,\u201d which we know is false, and negated it by saying \u201cIt is not true for every <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value that <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>.\u201d <\/p>\n<p>Symbolically, we originally said <span style=\"font-style:normal; font-size:90%\">\\(\\forall xP(x)\\)<\/span> , which is false. It can be negated by saying <\/p>\n<ul>\n<li><span style=\"font-style:normal; font-size:90%\">\\(\u00ac\\exists xP(x)\\)<\/span> (There exists an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span> such that <span style=\"font-style:normal; font-size:90%\">\\(x^{2}\\)<\/span> is not equal to 9) or<\/li>\n<li><span style=\"font-style:normal; font-size:90%\">\\(\\forall x\u00acP(x)\\)<\/span> (Not every <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value makes <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span> true)<\/li>\n<\/ul>\n<p>Again, both negations have the same meaning and are truth-equivalent (in this case, both true).<\/p>\n<p>Also, in this case, the term counterexample comes to mind. For every <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value, <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>. False, here\u2019s a counterexample: Let <span style=\"font-style:normal; font-size:90%\">\\(x=2\\)<\/span>, for instance.<\/p>\n<p>Let\u2019s try some practice: Evaluate each statement and show its negation has the opposite value.<\/p>\n<p>Every voter is at least 18 years old. This is true, according to the law. <br \/>\nNot every voter is at least 18 years old. And this is false, because the voting age is 18.<\/p>\n<p>Every composite number is even. False, some composite numbers are odd, such as 15.<br \/>\nNot every composite number is even. True.<\/p>\n<h2><span id=\"Multiple_Negations\" class=\"m-toc-anchor\"><\/span>Multiple Negations<\/h2>\n<p>\nStatements can be negated multiple times. Since one negation flips the truth value of a statement, a second negation flips it back. Let\u2019s expand our truth table:<\/p>\n<table class=\"ATable\" style=\"margin: auto;\">\n<thead>\n<tr>\n<th><strong>\\(P(x)\\)<\/strong><\/th>\n<th><strong>\\(\u00acP(x)\\)<\/strong><\/th>\n<th><strong>\\(\u00ac\u00acP(x)\\)<\/strong><\/th>\n<th><strong>\\(\u00ac\u00ac\u00acP(x)\\)<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>True<\/td>\n<td>False<\/td>\n<td>True<\/td>\n<td>False<\/td>\n<\/tr>\n<tr>\n<td>False<\/td>\n<td>True<\/td>\n<td>False<\/td>\n<td>True<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nTherefore, saying \u201cThere exists an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value such that<span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>,\u201d is truth equivalent to \u201cThere does not not exist an <span style=\"font-style:normal; font-size:90%\">\\(x\\)<\/span>-value such that <span style=\"font-style:normal; font-size:90%\">\\(x^{2}=9\\)<\/span>.\u201d  <span style=\"font-style:normal; font-size:90%\">\\(\\exists xP(x)=\u00ac\u00ac\\exists xP(x)\\)<\/span>.<\/p>\n<p>Now I want you to try some on your own! Pause the video and complete the chart.<\/p>\n<table class=\"ATable\" style=\"margin: auto; width: 90%;\">\n<thead>\n<tr>\n<th><strong>Statement<\/strong><\/th>\n<th><strong>Symbolic<\/strong><\/th>\n<th><strong>Truth<\/strong><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>All integers are rational numbers.<\/td>\n<td>\\(\\forall xP(x)\\)<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>Not all integers are rational numbers.<\/td>\n<td>\\(\u00ac\\forall xP(x)\\)<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>It is not the case that not all integers are rational numbers.<\/td>\n<td>\\(\u00ac\u00ac\\forall xP(x)\\)<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>There exist integers that are not rational numbers.<\/td>\n<td>\\(\\exists x\u00acP(x)\\)<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>It is false that it is not the case that not all integers are rational numbers.<\/td>\n<td>\\(\u00ac\u00ac\u00ac\\forall xP(x)\\)<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>There do not exist integers that are rational numbers.<\/td>\n<td>\\(\u00ac\\exists xP(x)\\)<\/td>\n<td>F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nI hope that this video increased your understanding of negation! Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Negation_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Negation Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nLet \\(\\exists xP(x)\\) be the statement \\(\\exists x:x^2=16\\). Which of the following is the statement for \\(\\lnot\\exists xP(x)\\) and its truth value? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\">There does not exist an \\(x\\)-value such that \\(x^2=16\\), which is a false statement.<\/div><div class=\"PQ\"  id=\"PQ-1-2\">There does not exist an \\(x\\)-value such that \\(x^2=16\\), which is a true statement.<\/div><div class=\"PQ\"  id=\"PQ-1-3\">There exists an \\(x\\)-value such that \\(x^2\\neq16\\), which is a false statement.<\/div><div class=\"PQ\"  id=\"PQ-1-4\">There does not exist an \\(x\\)-value such that \\(x^2\\neq16\\), which is a true statement.<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The negation of an existential statement is \\(\\lnot \\exists x \\, P(x)\\),which means: \u201cThere does not exist an \\(x\\)-value such that \\(x^2 = 16\\).\u201d<\/p>\n<p>To determine the truth value, note that \\(\\exists x : x^2 = 16\\) is true because \\(x = 4\\) (and also \\(x = -4\\)) satisfies the equation.<\/p>\n<p>Since the original statement is true, its negation must be false.<\/p>\n<p>Therefore, the correct answer choice is the one stating that \u201cThere does not exist an \\(x\\) such that \\(x^2 = 16\\),\u201d and that this statement is false.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nLet \\(\\forall xP(x)\\) be the statement \u201cFor every value of \\(x\\), \\(x^2=25\\)\u201d, which is a false statement. Which of the following is the symbolic negation for the statement, \u201cIt is not the case that not every value of \\(x\\) makes \\(x^2=25\\),\u201d that also has the same truth value of \\(\\forall xP(x)\\)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">\\(\\lnot\\forall xP(x)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-2\">\\(\\lnot\\exists xP(x)\\)<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-3\">\\(\\lnot\\lnot\\forall xP(x)\\)<\/div><div class=\"PQ\"  id=\"PQ-2-4\">\\(\\lnot\\lnot\\exists xP(x)\\)<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>For the symbolic statement \\(\\forall xP(x)\\), \\(\\forall x\\) means \u201cFor every value of \\(x\\),\u201d and \\(P(x)\\) is \\(x^2=25\\). We are told this is a false statement.<\/p>\n<p>When a statement is negated, its truth value is the opposite of what it was. Symbolically negating our statement once, we can use \\(\\lnot\\forall xP(x)\\), which is true since it can be stated as, \u201cNot every value of \\(x\\) makes \\(x^2=25\\).\u201d Use \\(x=4\\) to show that the square of not every \\(x\\)-value equals \\(25\\) because \\(4^2=16\\neq25\\).<\/p>\n<p>We can negate \\(\\lnot\\forall xP(x)\\), using \\(\\lnot\\lnot\\forall xP(x)\\), which becomes an equivalent false statement to \\(\\forall xP(x)\\). It can be stated as, \u201cIt is not the case that not every value of \\(x\\) makes \\(x^2=25\\).\u201d Since we saw from above that \u201cNot every value of \\(x\\) makes \\(x^2=25\\)\u201d was true, its negation must be false so, \\(\\lnot\\lnot\\forall xP(x)\\) has an equivalent truth statement using \\(P\\left(x\\right)\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nLet \\(\\forall xP(x)\\) be the statement, \u201cAll even numbers are integers.\u201d Which of the following is the statement for \\(\\exists x\\lnot P\\left(x\\right)\\) and its truth value? <\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">There does not exist an even number that is an integer, which is a false statement.<\/div><div class=\"PQ\"  id=\"PQ-3-2\">There does not exist an even number that is not an integer, which is a true statement.<\/div><div class=\"PQ\"  id=\"PQ-3-3\">There exists an even number that is not an integer, which is a true statement.<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-4\">There exists an even number that is not an integer, which is a false statement.<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The symbolic statement \\(\\exists x\\lnot P(x)\\) means to negate the conclusive portion of the statement \u201care integers.\u201d When a statement is negated, its truth value is the opposite of what it was. Negations often use the word \u201cnot\u201d when they are stated. Additionally, the symbol \\(\\exists\\) means that there exists. Since our statement is \u201cAll even numbers are integers\u201d, our negated statement, \\(\\exists x\\lnot P\\left(x\\right)\\), should state \u201cThere exists an even number that is not an integer.\u201d<\/p>\n<p>The set of integers contains the numbers \\(\\{&#8230;,-3,-2,-1,0,1,2,3,&#8230;\\}\\). The set of even numbers contains the numbers \\(\\{&#8230;,-6,-4,-2,0,2,4,6,&#8230;\\}\\) which is a subset of the set of integers. Therefore, all even numbers are also integers. This means the statement, \u201cAll even numbers are integers\u201d is a true statement. Thus, the negation \\(\\exists x\\lnot P(x)\\) must be false. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe statement, \u201cThe minimum voting age for all voters in the United States is 18 years old,\u201d is a true statement. What is the negation of the statement and its truth value?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">The minimum voting age for all voters in the United States is not 18 years old, which is a true statement.<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-2\">The minimum voting age for all voters in the United States is not 18 years old, which is a false statement.<\/div><div class=\"PQ\"  id=\"PQ-4-3\">There exists a voter in the United States whose minimum voting age is not 18 years old, which is a false statement.<\/div><div class=\"PQ\"  id=\"PQ-4-4\">There does not exist a voter in the United States whose minimum voting age is not 18 years old, which is a false statement.<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>When a statement is negated, its truth value is the opposite of what it was. We can negate the predicate of a statement by using the word \u201cnot\u201d when stating it. Since our predicate states \u201cis 18 years old,\u201d our negated statement will be \u201cThe minimum voting age of all voters in the United States is not 18 years old.\u201d <\/p>\n<p>Since it is given that the original statement is true, the negation of the original statement must be a false statement. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nThe statement, &#8220;Only one of my neighbors has exactly two dogs,&#8221; is a true statement. What is the negation of the statement?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">None of my neighbors do not have exactly two dogs. <\/div><div class=\"PQ\"  id=\"PQ-5-2\">There is at least one of my neighbors that does not have exactly two dogs.<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-3\">All of my neighbors do not have exactly two dogs.<\/div><div class=\"PQ\"  id=\"PQ-5-4\">Not all of my neighbors have exactly two dogs. <\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>When a statement is negated, its truth value is the opposite of what it was. We can negate the predicate of a statement by using the word \u201cnot\u201d or \u201cdoes not\u201d when stating it. Since our predicate states, \u201chas exactly two dogs\u201d we can write its negation as, \u201cdoes not have exactly two dogs.\u201d<\/p>\n<p>Additionally, if you only have two neighbors, then the negated statement, \u201cOnly one of my neighbors does not have exactly two dogs,\u201d is still a true statement since it implies the other neighbor does have exactly two dogs. The negated statement needs to be false. To make the negated statement false, we can change the initial part of the negation, \u201cOnly one of my neighbors\u201d to \u201cAll of my neighbors.\u201d So, the negation becomes, \u201cAll of my neighbors do not have exactly two dogs,\u201d which is a false statement. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/discrete-math\/\">Return to Discrete Math Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Discrete Math Videos<\/p>\n","protected":false},"author":1,"featured_media":100798,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-86455","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-discrete-math-videos","7":"page_category-proof-videos","8":"page_category-video-pages-for-study-course-sidebar-ad","9":"page_type-video","10":"content_type-practice-questions","11":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/86455","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\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=86455"}],"version-history":[{"count":7,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/86455\/revisions"}],"predecessor-version":[{"id":280289,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/86455\/revisions\/280289"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100798"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=86455"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}