{"id":168581,"date":"2023-02-07T13:19:49","date_gmt":"2023-02-07T19:19:49","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=168581"},"modified":"2025-12-11T12:12:54","modified_gmt":"2025-12-11T18:12:54","slug":"introduction-to-logical-reasoning","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/introduction-to-logical-reasoning\/","title":{"rendered":"Introduction to Logical Reasoning"},"content":{"rendered":"<h1>Introduction to Logical Reasoning<\/h1>\n\n\t\t\t<div id=\"mmDeferVideoEncompass_LWN2t_ioQqw\" 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_LWN2t_ioQqw\" data-source-videoID=\"LWN2t_ioQqw\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Introduction to Logical Reasoning Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Introduction to Logical Reasoning\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_LWN2t_ioQqw:hover {cursor:pointer;} img#videoThumbnailImage_LWN2t_ioQqw {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/02\/1796-thumb-final-2-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_LWN2t_ioQqw\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_LWN2t_ioQqw\" 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 Logical Reasoning\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_LWN2t_ioQqw\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_LWN2t_ioQqw\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_LWN2t_ioQqw\").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<h2>Logical Reasoning and Proofs<\/h2>\n<p>\nMuch of theoretical mathematics has to do with writing proofs to connect known truths to new findings. To help you write strong proofs, we\u2019re going to talk about logical reasoning, the foundation on which all valid proofs are built.<\/p>\n<p>One key element of proofs is the premises which are used in them. <strong>Premises<\/strong> are statements which are already assumed to be true, and from these true statements, new conclusions may be derived. For example, we already know that the interior angles of a triangle must add up to 180\u00b0. This fact may be regarded as a premise.<\/p>\n<p>Proofs as a whole may be referred to as arguments. An <strong>argument<\/strong> is a collection of premises followed by a conclusion. If the premises used in an argument are all true, and if the conclusion follows logically from the premises, then the argument is considered valid and the proof is complete.<\/p>\n<p>The individual sentences which compose a proof are called <strong>statements<\/strong> and may be either simple statements or compound statements. Statements must have a truth value, meaning they must be able to be proven true or false. \u201cThe sky is blue\u201d is a statement because it can be proven true. \u201cThere are a lot of clouds in the sky\u201d is not considered a statement because \u201ca lot\u201d is not specifically measurable. I may think there are a lot of clouds in the sky but you may disagree. So, the statement cannot be proven true or false. Opinions are never considered statements in math.<\/p>\n<p><strong>Simple statements<\/strong> are straightforward and are intended to convey one thought. <strong>Compound statements<\/strong>, on the other hand, include one or more simple statements along with a logical operator. The five logical operators are negation, conjunction, disjunction, conditional, and biconditional. If that all sounded like gibberish to you, there\u2019s no need to worry. Let\u2019s clear any confusion by going over what each of these mean.<\/p>\n<p>The <a href=\"https:\/\/www.mometrix.com\/academy\/negation\/\" style=\"text-decoration: underline;\"><strong>negation<\/strong><\/a> operator is denoted with a tilde, or squiggle, that precedes the simple statement, and its action is to undo, or give the opposite, of that statement. For example, if we wanted to negate the statement, \u201cThere are no trucks in the parking lot,\u201d this would be like saying, \u201cIt is not true that there are no trucks in the parking lot.\u201d In other words, we could say, \u201cThere is at least one truck in the parking lot.\u201d To negate a statement is to claim that it is untrue.<\/p>\n<div class=\"examplesentence\">\\(\\sim (no\\text{ } trucks)\\rightarrow at\\text{ }least\\text{ }one\\text{ }truck\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>The second type of logical operator is <strong>conjunction<\/strong>, which essentially connects two simple statements with the word \u201cand.\u201d The symbol for conjunction is an upside-down \u201cV\u201d. So, if we wanted to say, \u201cThere are no trucks in the parking lot and there are no cars in the parking lot,\u201d we could write:<\/p>\n<div class=\"examplesentence\">\\((\\text{no trucks})\\vee (\\text{no cars})\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>This statement would be true only if the parking lot contained zero cars and zero trucks.<\/p>\n<p>Notice that the conjunction symbol is similar to the symbol we use for the <a href=\"https:\/\/www.mometrix.com\/academy\/intro-to-set-theory\/\" style=\"text-decoration: underline;\"><strong>intersections of sets<\/strong><\/a>. In fact, a conjunction may be thought of as a type of intersection, because we use the word \u201cand\u201d to show that two statements (like sets) are fulfilled simultaneously.<\/p>\n<div class=\"examplesentence\">\\(\\wedge\\) \\(\\text{       }\\)\\(\\cap\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>The third logical operator is <strong>disjunction<\/strong>, which is denoted with a V-shaped symbol, and means \u201cor.\u201d If you guessed that disjunction is quite like the union of sets, you would be quite right! Disjunctions are satisfied when at least one of the simple statements given is true. For example, if we say, \u201cThere are no trucks in the parking lot or there are no cars in the parking lot,\u201d then we are correct as long as one or both of those are true. We are only wrong if both simple statements are false.<\/p>\n<div class=\"examplesentence\">\\((\\text{no trucks}) V (\\text{no cars})\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>The fourth logical operator is the <strong>conditional<\/strong>, which we can think of as an \u201cif, then\u201d statement. For example, the statement, \u201cIf it is a holiday, then there is no school\u201d is a conditional and can be denoted by writing a left-to-right arrow between the two simple statements.<\/p>\n<div class=\"examplesentence\">\\(\\text{Holiday} \\rightarrow \\text{no school}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>This arrow is usually pronounced using the word \u201cimplies.\u201d Conditionals are only untrue whenever the first statement is true but the second is false. For example, this conditional is true whenever there is a holiday and school is not being held for the day. The conditional is also true whenever there is no holiday and there is school. And even on days when there is no holiday, but also no school, this statement is still true\u2014it hasn\u2019t become a lie just because there is no holiday. The conditional only becomes false when there is a holiday and there is school.<\/p>\n<p>Conditionals only flow from one way, left to right, but the fifth logical operator, the <strong>biconditional<\/strong>, can be read forwards and backwards. Biconditionals usually involve the phrase \u201cif and only if,\u201d and are denoted by a two-sided arrow. For example, the statement, \u201cA shape is a triangle if and only if it is a three-sided polygon,\u201d is a biconditional statement. We would get the same information by reading it backwards: \u201cA shape is a three-sided polygon if and only if it is a triangle.\u201d Biconditional statements are true and satisfied whenever both simple statements are true, or both are false. If one is true and the other is false, then the biconditional is invalid.<\/p>\n<p>A couple final concepts that will help you in logic problems and proofs are existential and universal quantifiers. These names may sound a bit lofty, but in reality they aren\u2019t so bad. These quantifiers are simply tools for expressing how many objects you\u2019re talking about. Earlier, we said that the statement, \u201cThere are no trucks in the parking lot\u201d could be negated if there was at least one truck in the parking lot. If we did find a truck, we could say, \u201cThere is a truck in the parking lot.\u201d Whether we find one or one-hundred trucks, the initial statement is invalidated. In logic, we use the <strong>existential quantifier<\/strong>, a backwards-E symbol, as a shorthand for the words \u201cthere is.\u201d<\/p>\n<div class=\"examplesentence\">\\(\\exists (\\text{ a truck})\\)<\/div>\n<p>\n&nbsp;<\/p>\n<p>This symbol can also be pronounced with the words \u201cthere exists\u201d a truck, or \u201cthere is some\u201d truck.<\/p>\n<p>On the other hand, if we searched the parking lot and found zero trucks, we could say, \u201cFor all of the vehicles in the parking lot, none of them are trucks.\u201d The words \u201cfor all\u201d are used for a <strong>universal quantifier<\/strong>, and are written in shorthand by mathematicians as an upside-down \u201cA.\u201d<\/p>\n<div class=\"examplesentence\">\\(\\forall\\) vehicles, none are trucks<\/div>\n<p>\n&nbsp;<\/p>\n<p>\u201cFor all\u201d can also be pronounced as \u201cfor every\u201d or \u201cfor each.\u201d<\/p>\n<p>It is even possible to use <em>both<\/em> of the quantifiers in a sentence, and mathematicians frequently do. For example, we could say, \u201cFor all gloves, there is one thumb,\u201d or \u201cThere exists some house for which all rooms are painted pink.\u201d<\/p>\n<div class=\"examplesentence\">\\( \\forall \\text{ gloves},  \\exists \\text{ a thumb}\\)<\/div>\n<p>\n&nbsp;<\/p>\n<div class=\"examplesentence\">\\(\\exists\\) some house such that \\(\\forall\\) rooms, they are pink<\/div>\n<p>\n&nbsp;<\/p>\n<p>We\u2019ve now discussed that proofs are formed using a sequence of statements, all connected by a flow of logical thought. Many statements are compound in nature, involving one or more simple statements and a logical operator. These operators include negation, conjunction, disjunction, conditional, and biconditional. Additionally, the existential quantifier \u201cthere exists\u201d and the universal quantifier \u201cfor all\u201d can be very helpful in building successful proofs. With these tools, you\u2019re now ready to try some proof and logic problems on your own!<\/p>\n<p>I hope that 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 Logical Reasoning Return to Discrete Math Videos<\/p>\n","protected":false},"author":22,"featured_media":168701,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-168581","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\/168581","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=168581"}],"version-history":[{"count":3,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/168581\/revisions"}],"predecessor-version":[{"id":184361,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/168581\/revisions\/184361"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/168701"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=168581"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}