{"id":3278,"date":"2023-05-23T18:08:07","date_gmt":"2023-05-23T18:08:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=3278"},"modified":"2024-10-18T20:50:14","modified_gmt":"2024-10-18T20:50:14","slug":"logic-basics-fresh-take","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/logic-basics-fresh-take\/","title":{"raw":"Logic Basics: Fresh Take","rendered":"Logic Basics: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Combine sets using Boolean logic and proper notation<\/li>\r\n\t<li>Create and interpret expressions using statements and conditionals<\/li>\r\n\t<li>Construct and analyze truth tables for complex statements or conditionals<\/li>\r\n\t<li>Determine the logical equivalence between two statements<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Boolean Logic<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p><strong>Boolean logic<\/strong>, named after mathematician George Boole, is a system of logic that deals with binary variables (variables that can only take on one of two values, typically denoted as true or false). In the context of set theory, Boolean logic can be used to combine and manipulate sets.<\/p>\r\n<p>The primary operations in Boolean logic are \"and\" (corresponding to intersection in set theory), \"or\" (corresponding to union), and \"not\" (corresponding to complement). Here's how they work in the context of sets:<\/p>\r\n<ul>\r\n\t<li><strong>And (Intersection)<\/strong>: The intersection of two sets [latex]A[\/latex] and [latex]B[\/latex], represented as [latex]A \\cap B[\/latex], is the set of elements that are in both [latex]A[\/latex] and [latex]B[\/latex]. If we consider sets as conditions to be satisfied, the intersection represents the condition where both conditions [latex]A[\/latex] and [latex]B[\/latex] are true.<\/li>\r\n\t<li><strong>Or (Union)<\/strong>: The union of two sets [latex]A[\/latex] and [latex]B[\/latex], represented as [latex]A \\cup B[\/latex], is the set of elements that are in [latex]A[\/latex], in [latex]B[\/latex], or in both. In terms of conditions, the union represents the condition where either condition [latex]A[\/latex] or condition [latex]B[\/latex] or both are true.<\/li>\r\n\t<li><strong>Not (Complement)<\/strong>: The complement of a set [latex]A[\/latex], represented as [latex]\\bar{A}[\/latex] or [latex]A'[\/latex], is the set of all elements that are not in [latex]A[\/latex]. In terms of conditions, the complement represents the negation of condition A.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Describe the numbers that meet the condition: even and less than [latex]10[\/latex] and greater than [latex]0[\/latex].<br \/>\r\n[reveal-answer q=\"670488\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"670488\"]The numbers that satisfy all three requirements are [latex]\\{2, 4, 6, 8\\}[\/latex].[\/hidden-answer]<\/section>\r\n<section class=\"textbox watchIt\">\r\n<p><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=12421117&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=KrR7D58Onzw&amp;video_target=tpm-plugin-sh2v1jhj-KrR7D58Onzw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Pirates+vs+Ninjas_+Understanding+Boolean+Operators.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cBoolean Operators: Pirates vs. Ninjas\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Quantified Statements<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>In mathematics, a <strong>quantifier <\/strong>is a symbol or word used to express the extent to which a predicate (or proposition) is true over a range of values. There are two main types of quantifiers in mathematics: universal quantifiers and existential quantifiers.<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>The <strong>universal quantifier<\/strong> is typically symbolized by [latex]\\forall[\/latex] (an inverted letter 'A'), which stands for \"for all\" or \"for every.\" When we use the universal quantifier, we are saying that a certain property or condition holds for all members of a specific set.<\/li>\r\n\t<li>The <strong>existential quantifier<\/strong> is symbolized by [latex]\\exists[\/latex] (a backwards letter 'E'), which stands for \"there exists.\" When we use the existential quantifier, we are saying that there is at least one member of a set that satisfies a certain property or condition.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">Suppose your friend says \u201cOne of these six cartons of milk is leaking.\u201d What is the minimum amount of evidence you would need to prove your friend wrong?[reveal-answer q=\"877482\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"877482\"]<br \/>\r\nIn this case, you would need to check <u>all<\/u> six cartons and show that none of them is leaking. You cannot disprove your friend\u2019s statement by checking only one of the cartons.<br \/>\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205777&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=GJpezCUMOxA&amp;video_target=tpm-plugin-0nmyug9f-GJpezCUMOxA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Universal+and+Existential+Quantifiers+%E2%88%80+For+All+and+%E2%88%83+There+Exists.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUniversal and Existential Quantifiers, \u2200 \"For All\" and \u2203 \"There Exists\"\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h3>Negating Quantified Statements<\/h3>\r\n<p>In mathematical logic, the process of negating a quantified statement involves changing the quantifier and negating the predicate. The negation of a universally quantified statement results in an existentially quantified statement and vice versa. The exact rules for negating quantified statements are as follows:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li><strong>Negation of Universal Quantification<\/strong>: If we have a universally quantified statement such as [latex]\\forall x, P(x)[\/latex] (read as: for all [latex]x[\/latex], [latex]P(x)[\/latex] is true), the negation of this statement is [latex]\\exists x, \\neg P(x)[\/latex] (read as: there exists an [latex]x[\/latex] such that [latex]P(x)[\/latex] is not true). Essentially, we are saying that there is at least one case where the proposition does not hold.<\/li>\r\n\t<li><strong>Negation of Existential Quantification<\/strong>: If we have an existentially quantified statement such as [latex]\\exists x, P(x)[\/latex] (read as: there exists an [latex]x[\/latex] such that [latex]P(x)[\/latex] is true), the negation of this statement is [latex]\\forall x, \\neg P(x)[\/latex] (read as: for all [latex]x[\/latex], [latex]P(x)[\/latex] is not true). In other words, we are saying that there is no case where the proposition holds.<\/li>\r\n<\/ol>\r\n<section class=\"textbox example\">Write the negation of this statement: \u201cThere are no prime numbers that are even.\u201d[reveal-answer q=\"877481\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"877481\"]<br \/>\r\nThe negation is \u201cAt least one prime number is even.\u201d<br \/>\r\n[\/hidden-answer]<\/section>\r\n<h2>Conditional Statements<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li>A <strong>statement <\/strong>is a declarative sentence that is either true or false but not both. For instance, the statement \"[latex]p[\/latex]: The sky is blue\" is a statement that can be true or false.<\/li>\r\n\t<li>A <strong>conditional statement<\/strong> is a compound statement of the form \"if [latex]p[\/latex] then [latex]q[\/latex]\" (denoted as [latex]p \u2192 q[\/latex]), where [latex]p[\/latex] and [latex]q[\/latex] are statements.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205778&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=C64fHC1hCxA&amp;video_target=tpm-plugin-p46zx47r-C64fHC1hCxA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Conditional+Statements+Converse+Statements+%7C+Mathematical+Reasoning+%7C+Dont+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConditional Statements &amp; Converse Statements | Mathematical Reasoning | Don't Memorise\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h3>Conditional Statements and Excel<\/h3>\r\n<p>Excel provides powerful tools for implementing conditional logic, enabling you to manipulate and analyze data based on specific conditions.<\/p>\r\n<p><strong>IF Statements<\/strong>: The IF function is a fundamental tool for creating conditional statements in Excel. It tests a condition and returns one result if the condition is true and another if it's false. The basic syntax is IF(logical_test, value_if_true, value_if_false). For example, =IF([latex]A1&gt;5[\/latex],\"Greater\",\"Smaller\") would return \"Greater\" if the value in cell [latex]A1[\/latex] is greater than [latex]5[\/latex], and \"Smaller\" otherwise.<\/p>\r\n<p><strong>AND, OR, NOT Functions<\/strong>: You can use the AND, OR, and NOT functions to create more complex conditions. The AND function returns TRUE if all its arguments are true, the OR function returns TRUE if any of its arguments are true, and the NOT function reverses the value of its argument.<\/p>\r\n<section class=\"textbox example\">Write an Excel command that will create the condition \u201c[latex]A1 &lt; 3000[\/latex] and [latex]A1 &gt; 100[\/latex]\u201d.<br \/>\r\n[reveal-answer q=\"870468\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"870468\"]Enter \u201c[latex]\\text{AND}(A1&lt;3000, A1&gt;100)[\/latex]\u201d.[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">An accountant needs to withhold [latex]15\\%[\/latex] of income for taxes if the income is below [latex]$30,000[\/latex], and [latex]20\\%[\/latex] of income if the income is [latex]$30,000[\/latex] or more.\u00a0Write an Excel expression that would calculate the amount to withhold.<br \/>\r\n[reveal-answer q=\"586943\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"586943\"]Our conditional needs to compare the value to [latex]30,000[\/latex]. If the income is less than [latex]30,000[\/latex], we need to calculate [latex]15\\%[\/latex] of the income: [latex]0.15*\\text{income}[\/latex]. If the income is more than [latex]30,000[\/latex], we need to calculate [latex]20\\%[\/latex] of the income: [latex]0.20*\\text{income}[\/latex].In words we could write \u201cIf [latex]\\text{income} &lt; 30,000[\/latex], then multiply by [latex]0.15[\/latex], otherwise multiply by [latex]0.20[\/latex]\u201d. In Excel, we would write:<br \/>\r\n<center>[latex]=\\text{IF}(A1&lt;30000, 0.15*A1, 0.20*A1)[\/latex]<\/center><br \/>\r\n[\/hidden-answer]<\/section>\r\n<h2>Truth Tables<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>A <strong> truth table<\/strong> is a table showing what the resulting truth value of a complex statement is for all the possible truth values for the simple statements.<\/p>\r\n<p>The symbol [latex]\\wedge[\/latex] is used for <em>and<\/em>: [latex]A[\/latex] and [latex]B[\/latex] is notated [latex]A\\wedge{B}[\/latex] .<\/p>\r\n<p>The symbol [latex]\\vee[\/latex] is used for <em>or<\/em>: [latex]A[\/latex] or [latex]B[\/latex] is notated [latex]A\\vee{B}[\/latex] .<\/p>\r\n<p>The symbol [latex]\\sim[\/latex] is used for <em>not<\/em>: not [latex]A[\/latex] is notated [latex]\\sim{A}[\/latex] .<\/p>\r\n<\/div>\r\n<p>To create a truth table, you need to:<\/p>\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Identify the Propositions: Start by identifying the individual propositions or statements that make up your complex statement.<\/li>\r\n\t<li>Create the Table: Draw a table with one column for each proposition and one for the complex statement. The table should have as many rows as there are possible combinations of truth values for the propositions - this is typically [latex]2^n[\/latex], where [latex]n[\/latex] is the number of propositions.<\/li>\r\n\t<li>Fill in the Truth Values: In the proposition columns, fill in all possible combinations of truth and falsehood. Start with alternating truth and falsehood in the first column, then alternate pairs of truths and falsehoods in the second, and so on.<\/li>\r\n\t<li>Evaluate the Complex Statement: For each row, evaluate the truth or falsehood of the complex statement based on the truth values of the propositions in that row.<\/li>\r\n<\/ol>\r\n<p>For example, consider the complex statement [latex]P \\text{ AND } Q[\/latex] (written as [latex]P \u2227 Q[\/latex] in logic). The truth table would look like this:<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th style=\"text-align: center;\">[latex]P[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]Q[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]P \u2227 Q[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<section class=\"textbox example\">Translate \u201cWe have carrots or we will not make soup\u201d into symbols. Let [latex]C[\/latex] represent \u201cwe have carrots\u201d and let [latex]S[\/latex] represent \u201cwe will make soup.\u201d<br \/>\r\n[reveal-answer q=\"586949\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"586949\"]<br \/>\r\n[latex]C \\vee \\sim S[\/latex]<br \/>\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">Create a truth table for this statement: [latex]\\sim A \\wedge B[\/latex] .<br \/>\r\n[reveal-answer q=\"550658\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"550658\"]\r\n\r\n<table width=\"40%\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\">[latex]A[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]B[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]\\sim A[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]\\sim A \\wedge B[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Create a truth table for this statement: [latex](\\sim A \\wedge B) \\vee \\sim B[\/latex] .<br \/>\r\n[reveal-answer q=\"550657\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"550657\"]\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: center;\">[latex]A[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]B[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\\sim A[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\\sim A \\wedge B [\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex]\\sim B[\/latex]<\/td>\r\n<td style=\"text-align: center;\">[latex](\\sim A \\wedge B) \\vee \\sim B[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205779&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=7mulE-zramc&amp;video_target=tpm-plugin-wzyf5q03-7mulE-zramc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Intro+to+Truth+Tables+%7C+Negotiation+Conjunction+and+Disjunction.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntro to Truth Tables | Negotiation, Conjunction, and Disjunction\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<h2>Implications<\/h2>\r\n<p>In logic, an implication, also known as a conditional statement, is a type of compound statement that consists of two parts, namely an antecedent ([latex]P[\/latex]) and a consequent ([latex]Q[\/latex]). The implication is often expressed as \"If [latex]P[\/latex], then [latex]Q[\/latex]\" or \"[latex]P[\/latex] implies [latex]Q[\/latex]\" and is denoted as \"[latex]P \u2192 Q[\/latex]\".<\/p>\r\n<p>The implication is said to be true except in the case where [latex]P[\/latex] is true and [latex]Q[\/latex] is false. Here's a truth table to illustrate:<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th style=\"text-align: center;\">[latex]P[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]Q[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]P \u2192 Q[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>An important concept related to implications is the converse, inverse, and contrapositive:<\/p>\r\n<ul>\r\n\t<li>The converse of an implication \"[latex]P \u2192 Q[\/latex]\" is \"[latex]Q \u2192 P[\/latex]\". The truth of an implication does not guarantee the truth of its converse. They can be true or false independently of each other.<\/li>\r\n\t<li>The inverse of an implication \"[latex]P \u2192 Q[\/latex]\" is \"[latex]\u00acP \u2192 \u00acQ[\/latex]\". Like the converse, the truth of an implication does not determine the truth of its inverse.<\/li>\r\n\t<li>The contrapositive of an implication \"[latex]P \u2192 Q[\/latex]\" is \"[latex]\u00acQ \u2192 \u00acP[\/latex]\". In logic, an implication and its contrapositive always share the same truth value. If the implication is true, then its contrapositive is true, and vice versa.<\/li>\r\n<\/ul>\r\n<h2>Equivalence<\/h2>\r\n<p>In the realm of logic and mathematics, equivalence refers to a relationship between two logical statements where both statements are either true or false simultaneously. This means if we have two statements, [latex]P[\/latex] and [latex]Q[\/latex], they are considered equivalent, denoted as [latex]P \\leftrightarrow Q[\/latex], if and only if they are both true or both false.<\/p>\r\n<p>Let's illustrate this with a truth table:<\/p>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th style=\"text-align: center;\">[latex]P[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]Q[\/latex]<\/th>\r\n<th style=\"text-align: center;\">[latex]P \u2194 Q[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">F<\/td>\r\n<td style=\"text-align: center;\">T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>&nbsp;<\/p>\r\n<p>As the table shows, the biconditional statement ([latex]P \\leftrightarrow Q[\/latex]) is true only when both [latex]P[\/latex] and [latex]Q[\/latex] share the same truth value. If [latex]P[\/latex] and [latex]Q[\/latex] are both true or both false, then [latex]P \\leftrightarrow Q[\/latex] is true. However, if [latex]P[\/latex] and [latex]Q[\/latex] have different truth values, then [latex]P \\leftrightarrow Q[\/latex] is false.<\/p>\r\n<h3>DeMorgan\u2019s Laws<\/h3>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<p>In the field of logic and mathematics, <strong>DeMorgan\u2019s Laws<\/strong> are fundamental rules that dictate how logical operations of conjunction (AND, represented as [latex]\\land[\/latex]) and disjunction (OR, represented as [latex]\\lor[\/latex]) interact with the negation operation (NOT, represented as [latex]\\neg[\/latex]). DeMorgan\u2019s Laws allow us to simplify complex logical expressions and provide a method for moving a negation operator across a conjunction or disjunction operator.<\/p>\r\n<p>DeMorgan\u2019s Laws can be stated as follows:<\/p>\r\n<ul>\r\n\t<li>[latex]\\sim (P \\land Q) = \\sim P \\lor \\sim Q[\/latex]<\/li>\r\n\t<li>[latex]\\sim (P \\lor Q) = \\sim P \\land \\sim Q[\/latex]<\/li>\r\n<\/ul>\r\n<p>In plain English, these laws mean:<\/p>\r\n<ul>\r\n\t<li>The negation of a conjunction ([latex]P \\land Q[\/latex]) is equivalent to the disjunction of the negations ([latex]\\sim P \\lor \\sim Q[\/latex]).<\/li>\r\n\t<li>The negation of a disjunction ([latex]P \\lor Q[\/latex]) is equivalent to the conjunction of the negations ([latex]\\sim P \\land \\sim Q[\/latex]).<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox watchIt\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=10205780&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=tKnS3s8fOu4&amp;video_target=tpm-plugin-a3uc7zvr-tKnS3s8fOu4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><br \/>\r\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Logic+-+DeMorgans+Laws+of+Negation.html\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogic - DeMorgan's Laws of Negation\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Combine sets using Boolean logic and proper notation<\/li>\n<li>Create and interpret expressions using statements and conditionals<\/li>\n<li>Construct and analyze truth tables for complex statements or conditionals<\/li>\n<li>Determine the logical equivalence between two statements<\/li>\n<\/ul>\n<\/section>\n<h2>Boolean Logic<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Boolean logic<\/strong>, named after mathematician George Boole, is a system of logic that deals with binary variables (variables that can only take on one of two values, typically denoted as true or false). In the context of set theory, Boolean logic can be used to combine and manipulate sets.<\/p>\n<p>The primary operations in Boolean logic are &#8220;and&#8221; (corresponding to intersection in set theory), &#8220;or&#8221; (corresponding to union), and &#8220;not&#8221; (corresponding to complement). Here&#8217;s how they work in the context of sets:<\/p>\n<ul>\n<li><strong>And (Intersection)<\/strong>: The intersection of two sets [latex]A[\/latex] and [latex]B[\/latex], represented as [latex]A \\cap B[\/latex], is the set of elements that are in both [latex]A[\/latex] and [latex]B[\/latex]. If we consider sets as conditions to be satisfied, the intersection represents the condition where both conditions [latex]A[\/latex] and [latex]B[\/latex] are true.<\/li>\n<li><strong>Or (Union)<\/strong>: The union of two sets [latex]A[\/latex] and [latex]B[\/latex], represented as [latex]A \\cup B[\/latex], is the set of elements that are in [latex]A[\/latex], in [latex]B[\/latex], or in both. In terms of conditions, the union represents the condition where either condition [latex]A[\/latex] or condition [latex]B[\/latex] or both are true.<\/li>\n<li><strong>Not (Complement)<\/strong>: The complement of a set [latex]A[\/latex], represented as [latex]\\bar{A}[\/latex] or [latex]A'[\/latex], is the set of all elements that are not in [latex]A[\/latex]. In terms of conditions, the complement represents the negation of condition A.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Describe the numbers that meet the condition: even and less than [latex]10[\/latex] and greater than [latex]0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q670488\">Show Solution<\/button><\/p>\n<div id=\"q670488\" class=\"hidden-answer\" style=\"display: none\">The numbers that satisfy all three requirements are [latex]\\{2, 4, 6, 8\\}[\/latex].<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\">\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=12421117&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=KrR7D58Onzw&amp;video_target=tpm-plugin-sh2v1jhj-KrR7D58Onzw\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><span data-mce-type=\"bookmark\" style=\"display: inline-block; width: 0px; overflow: hidden; line-height: 0;\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Pirates+vs+Ninjas_+Understanding+Boolean+Operators.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cBoolean Operators: Pirates vs. Ninjas\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Quantified Statements<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>In mathematics, a <strong>quantifier <\/strong>is a symbol or word used to express the extent to which a predicate (or proposition) is true over a range of values. There are two main types of quantifiers in mathematics: universal quantifiers and existential quantifiers.<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>The <strong>universal quantifier<\/strong> is typically symbolized by [latex]\\forall[\/latex] (an inverted letter &#8216;A&#8217;), which stands for &#8220;for all&#8221; or &#8220;for every.&#8221; When we use the universal quantifier, we are saying that a certain property or condition holds for all members of a specific set.<\/li>\n<li>The <strong>existential quantifier<\/strong> is symbolized by [latex]\\exists[\/latex] (a backwards letter &#8216;E&#8217;), which stands for &#8220;there exists.&#8221; When we use the existential quantifier, we are saying that there is at least one member of a set that satisfies a certain property or condition.<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">Suppose your friend says \u201cOne of these six cartons of milk is leaking.\u201d What is the minimum amount of evidence you would need to prove your friend wrong?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q877482\">Show Solution<\/button><\/p>\n<div id=\"q877482\" class=\"hidden-answer\" style=\"display: none\">\nIn this case, you would need to check <u>all<\/u> six cartons and show that none of them is leaking. You cannot disprove your friend\u2019s statement by checking only one of the cartons.\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205777&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=GJpezCUMOxA&amp;video_target=tpm-plugin-0nmyug9f-GJpezCUMOxA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Universal+and+Existential+Quantifiers+%E2%88%80+For+All+and+%E2%88%83+There+Exists.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cUniversal and Existential Quantifiers, \u2200 &#8220;For All&#8221; and \u2203 &#8220;There Exists&#8221;\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Negating Quantified Statements<\/h3>\n<p>In mathematical logic, the process of negating a quantified statement involves changing the quantifier and negating the predicate. The negation of a universally quantified statement results in an existentially quantified statement and vice versa. The exact rules for negating quantified statements are as follows:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li><strong>Negation of Universal Quantification<\/strong>: If we have a universally quantified statement such as [latex]\\forall x, P(x)[\/latex] (read as: for all [latex]x[\/latex], [latex]P(x)[\/latex] is true), the negation of this statement is [latex]\\exists x, \\neg P(x)[\/latex] (read as: there exists an [latex]x[\/latex] such that [latex]P(x)[\/latex] is not true). Essentially, we are saying that there is at least one case where the proposition does not hold.<\/li>\n<li><strong>Negation of Existential Quantification<\/strong>: If we have an existentially quantified statement such as [latex]\\exists x, P(x)[\/latex] (read as: there exists an [latex]x[\/latex] such that [latex]P(x)[\/latex] is true), the negation of this statement is [latex]\\forall x, \\neg P(x)[\/latex] (read as: for all [latex]x[\/latex], [latex]P(x)[\/latex] is not true). In other words, we are saying that there is no case where the proposition holds.<\/li>\n<\/ol>\n<section class=\"textbox example\">Write the negation of this statement: \u201cThere are no prime numbers that are even.\u201d<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q877481\">Show Solution<\/button><\/p>\n<div id=\"q877481\" class=\"hidden-answer\" style=\"display: none\">\nThe negation is \u201cAt least one prime number is even.\u201d\n<\/div>\n<\/div>\n<\/section>\n<h2>Conditional Statements<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li>A <strong>statement <\/strong>is a declarative sentence that is either true or false but not both. For instance, the statement &#8220;[latex]p[\/latex]: The sky is blue&#8221; is a statement that can be true or false.<\/li>\n<li>A <strong>conditional statement<\/strong> is a compound statement of the form &#8220;if [latex]p[\/latex] then [latex]q[\/latex]&#8221; (denoted as [latex]p \u2192 q[\/latex]), where [latex]p[\/latex] and [latex]q[\/latex] are statements.<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205778&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=C64fHC1hCxA&amp;video_target=tpm-plugin-p46zx47r-C64fHC1hCxA\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Conditional+Statements+Converse+Statements+%7C+Mathematical+Reasoning+%7C+Dont+Memorise.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cConditional Statements &amp; Converse Statements | Mathematical Reasoning | Don&#8217;t Memorise\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Conditional Statements and Excel<\/h3>\n<p>Excel provides powerful tools for implementing conditional logic, enabling you to manipulate and analyze data based on specific conditions.<\/p>\n<p><strong>IF Statements<\/strong>: The IF function is a fundamental tool for creating conditional statements in Excel. It tests a condition and returns one result if the condition is true and another if it&#8217;s false. The basic syntax is IF(logical_test, value_if_true, value_if_false). For example, =IF([latex]A1>5[\/latex],&#8221;Greater&#8221;,&#8221;Smaller&#8221;) would return &#8220;Greater&#8221; if the value in cell [latex]A1[\/latex] is greater than [latex]5[\/latex], and &#8220;Smaller&#8221; otherwise.<\/p>\n<p><strong>AND, OR, NOT Functions<\/strong>: You can use the AND, OR, and NOT functions to create more complex conditions. The AND function returns TRUE if all its arguments are true, the OR function returns TRUE if any of its arguments are true, and the NOT function reverses the value of its argument.<\/p>\n<section class=\"textbox example\">Write an Excel command that will create the condition \u201c[latex]A1 < 3000[\/latex] and [latex]A1 > 100[\/latex]\u201d.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q870468\">Show Solution<\/button><\/p>\n<div id=\"q870468\" class=\"hidden-answer\" style=\"display: none\">Enter \u201c[latex]\\text{AND}(A1<3000, A1>100)[\/latex]\u201d.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">An accountant needs to withhold [latex]15\\%[\/latex] of income for taxes if the income is below [latex]$30,000[\/latex], and [latex]20\\%[\/latex] of income if the income is [latex]$30,000[\/latex] or more.\u00a0Write an Excel expression that would calculate the amount to withhold.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q586943\">Show Solution<\/button><\/p>\n<div id=\"q586943\" class=\"hidden-answer\" style=\"display: none\">Our conditional needs to compare the value to [latex]30,000[\/latex]. If the income is less than [latex]30,000[\/latex], we need to calculate [latex]15\\%[\/latex] of the income: [latex]0.15*\\text{income}[\/latex]. If the income is more than [latex]30,000[\/latex], we need to calculate [latex]20\\%[\/latex] of the income: [latex]0.20*\\text{income}[\/latex].In words we could write \u201cIf [latex]\\text{income} < 30,000[\/latex], then multiply by [latex]0.15[\/latex], otherwise multiply by [latex]0.20[\/latex]\u201d. In Excel, we would write:<\/p>\n<div style=\"text-align: center;\">[latex]=\\text{IF}(A1<30000, 0.15*A1, 0.20*A1)[\/latex]<\/div>\n<p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Truth Tables<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>A <strong> truth table<\/strong> is a table showing what the resulting truth value of a complex statement is for all the possible truth values for the simple statements.<\/p>\n<p>The symbol [latex]\\wedge[\/latex] is used for <em>and<\/em>: [latex]A[\/latex] and [latex]B[\/latex] is notated [latex]A\\wedge{B}[\/latex] .<\/p>\n<p>The symbol [latex]\\vee[\/latex] is used for <em>or<\/em>: [latex]A[\/latex] or [latex]B[\/latex] is notated [latex]A\\vee{B}[\/latex] .<\/p>\n<p>The symbol [latex]\\sim[\/latex] is used for <em>not<\/em>: not [latex]A[\/latex] is notated [latex]\\sim{A}[\/latex] .<\/p>\n<\/div>\n<p>To create a truth table, you need to:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Identify the Propositions: Start by identifying the individual propositions or statements that make up your complex statement.<\/li>\n<li>Create the Table: Draw a table with one column for each proposition and one for the complex statement. The table should have as many rows as there are possible combinations of truth values for the propositions &#8211; this is typically [latex]2^n[\/latex], where [latex]n[\/latex] is the number of propositions.<\/li>\n<li>Fill in the Truth Values: In the proposition columns, fill in all possible combinations of truth and falsehood. Start with alternating truth and falsehood in the first column, then alternate pairs of truths and falsehoods in the second, and so on.<\/li>\n<li>Evaluate the Complex Statement: For each row, evaluate the truth or falsehood of the complex statement based on the truth values of the propositions in that row.<\/li>\n<\/ol>\n<p>For example, consider the complex statement [latex]P \\text{ AND } Q[\/latex] (written as [latex]P \u2227 Q[\/latex] in logic). The truth table would look like this:<\/p>\n<table>\n<tbody>\n<tr>\n<th style=\"text-align: center;\">[latex]P[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]Q[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]P \u2227 Q[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<section class=\"textbox example\">Translate \u201cWe have carrots or we will not make soup\u201d into symbols. Let [latex]C[\/latex] represent \u201cwe have carrots\u201d and let [latex]S[\/latex] represent \u201cwe will make soup.\u201d<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q586949\">Show Solution<\/button><\/p>\n<div id=\"q586949\" class=\"hidden-answer\" style=\"display: none\">\n[latex]C \\vee \\sim S[\/latex]\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Create a truth table for this statement: [latex]\\sim A \\wedge B[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q550658\">Show Solution<\/button><\/p>\n<div id=\"q550658\" class=\"hidden-answer\" style=\"display: none\">\n<table style=\"width: 40%;\">\n<thead>\n<tr>\n<th style=\"text-align: center;\">[latex]A[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]B[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]\\sim A[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]\\sim A \\wedge B[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Create a truth table for this statement: [latex](\\sim A \\wedge B) \\vee \\sim B[\/latex] .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q550657\">Show Solution<\/button><\/p>\n<div id=\"q550657\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: center;\">[latex]A[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]B[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\sim A[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\sim A \\wedge B[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex]\\sim B[\/latex]<\/td>\n<td style=\"text-align: center;\">[latex](\\sim A \\wedge B) \\vee \\sim B[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205779&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=7mulE-zramc&amp;video_target=tpm-plugin-wzyf5q03-7mulE-zramc\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Intro+to+Truth+Tables+%7C+Negotiation+Conjunction+and+Disjunction.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIntro to Truth Tables | Negotiation, Conjunction, and Disjunction\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Implications<\/h2>\n<p>In logic, an implication, also known as a conditional statement, is a type of compound statement that consists of two parts, namely an antecedent ([latex]P[\/latex]) and a consequent ([latex]Q[\/latex]). The implication is often expressed as &#8220;If [latex]P[\/latex], then [latex]Q[\/latex]&#8221; or &#8220;[latex]P[\/latex] implies [latex]Q[\/latex]&#8221; and is denoted as &#8220;[latex]P \u2192 Q[\/latex]&#8220;.<\/p>\n<p>The implication is said to be true except in the case where [latex]P[\/latex] is true and [latex]Q[\/latex] is false. Here&#8217;s a truth table to illustrate:<\/p>\n<table>\n<tbody>\n<tr>\n<th style=\"text-align: center;\">[latex]P[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]Q[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]P \u2192 Q[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>An important concept related to implications is the converse, inverse, and contrapositive:<\/p>\n<ul>\n<li>The converse of an implication &#8220;[latex]P \u2192 Q[\/latex]&#8221; is &#8220;[latex]Q \u2192 P[\/latex]&#8220;. The truth of an implication does not guarantee the truth of its converse. They can be true or false independently of each other.<\/li>\n<li>The inverse of an implication &#8220;[latex]P \u2192 Q[\/latex]&#8221; is &#8220;[latex]\u00acP \u2192 \u00acQ[\/latex]&#8220;. Like the converse, the truth of an implication does not determine the truth of its inverse.<\/li>\n<li>The contrapositive of an implication &#8220;[latex]P \u2192 Q[\/latex]&#8221; is &#8220;[latex]\u00acQ \u2192 \u00acP[\/latex]&#8220;. In logic, an implication and its contrapositive always share the same truth value. If the implication is true, then its contrapositive is true, and vice versa.<\/li>\n<\/ul>\n<h2>Equivalence<\/h2>\n<p>In the realm of logic and mathematics, equivalence refers to a relationship between two logical statements where both statements are either true or false simultaneously. This means if we have two statements, [latex]P[\/latex] and [latex]Q[\/latex], they are considered equivalent, denoted as [latex]P \\leftrightarrow Q[\/latex], if and only if they are both true or both false.<\/p>\n<p>Let&#8217;s illustrate this with a truth table:<\/p>\n<table>\n<tbody>\n<tr>\n<th style=\"text-align: center;\">[latex]P[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]Q[\/latex]<\/th>\n<th style=\"text-align: center;\">[latex]P \u2194 Q[\/latex]<\/th>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">F<\/td>\n<td style=\"text-align: center;\">T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p>As the table shows, the biconditional statement ([latex]P \\leftrightarrow Q[\/latex]) is true only when both [latex]P[\/latex] and [latex]Q[\/latex] share the same truth value. If [latex]P[\/latex] and [latex]Q[\/latex] are both true or both false, then [latex]P \\leftrightarrow Q[\/latex] is true. However, if [latex]P[\/latex] and [latex]Q[\/latex] have different truth values, then [latex]P \\leftrightarrow Q[\/latex] is false.<\/p>\n<h3>DeMorgan\u2019s Laws<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>In the field of logic and mathematics, <strong>DeMorgan\u2019s Laws<\/strong> are fundamental rules that dictate how logical operations of conjunction (AND, represented as [latex]\\land[\/latex]) and disjunction (OR, represented as [latex]\\lor[\/latex]) interact with the negation operation (NOT, represented as [latex]\\neg[\/latex]). DeMorgan\u2019s Laws allow us to simplify complex logical expressions and provide a method for moving a negation operator across a conjunction or disjunction operator.<\/p>\n<p>DeMorgan\u2019s Laws can be stated as follows:<\/p>\n<ul>\n<li>[latex]\\sim (P \\land Q) = \\sim P \\lor \\sim Q[\/latex]<\/li>\n<li>[latex]\\sim (P \\lor Q) = \\sim P \\land \\sim Q[\/latex]<\/li>\n<\/ul>\n<p>In plain English, these laws mean:<\/p>\n<ul>\n<li>The negation of a conjunction ([latex]P \\land Q[\/latex]) is equivalent to the disjunction of the negations ([latex]\\sim P \\lor \\sim Q[\/latex]).<\/li>\n<li>The negation of a disjunction ([latex]P \\lor Q[\/latex]) is equivalent to the conjunction of the negations ([latex]\\sim P \\land \\sim Q[\/latex]).<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox watchIt\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=10205780&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=tKnS3s8fOu4&amp;video_target=tpm-plugin-a3uc7zvr-tKnS3s8fOu4\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Logic+-+DeMorgans+Laws+of+Negation.html\" target=\"_blank\" rel=\"noopener\">transcript for \u201cLogic &#8211; DeMorgan&#8217;s Laws of Negation\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":15,"menu_order":18,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":24,"module-header":"fresh_take","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3278"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":56,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3278\/revisions"}],"predecessor-version":[{"id":15075,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3278\/revisions\/15075"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/24"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/3278\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=3278"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=3278"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=3278"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=3278"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}