{"id":864,"date":"2016-12-19T20:54:23","date_gmt":"2016-12-19T20:54:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/?post_type=chapter&#038;p=864"},"modified":"2019-05-30T16:37:21","modified_gmt":"2019-05-30T16:37:21","slug":"truth-tables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/chapter\/truth-tables\/","title":{"raw":"Truth Tables","rendered":"Truth Tables"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Combine sets using Boolean logic, using proper notations<\/li>\r\n \t<li>Use statements and conditionals to write and interpret expressions<\/li>\r\n \t<li>Use a truth table to interpret complex statements or conditionals<\/li>\r\n \t<li>Write truth tables given a logical implication, and it\u2019s related\u00a0statements \u2013 converse, inverse, and contrapositive<\/li>\r\n \t<li>Determine whether two statements are logically equivalent<\/li>\r\n \t<li>Use DeMorgan\u2019s laws to define logical equivalences of a statement<\/li>\r\n<\/ul>\r\n<\/div>\r\nBecause complex Boolean statements can get tricky to think about, we can create a <strong>truth table<\/strong> to break the complex statement into\u00a0simple statements, and determine whether they are true or false. A table will help keep track of all the truth values of the simple statements that make up a complex statement, leading to an analysis of the full statement.\r\n<div class=\"textbox\">\r\n<h3>Truth Table<\/h3>\r\nA table showing what the resulting truth value of a complex statement is for all the possible truth values for the simple statements.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSuppose you\u2019re picking out a new couch, and your significant other says \u201cget a sectional <em>or<\/em> something with a chaise.\u201d Construct a truth table that describes the elements of the conditions of this statement and whether the conditions are met.\r\n[reveal-answer q=\"14714\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"14714\"]\r\n\r\nThis is a complex statement made of two simpler conditions: \u201cis a sectional,\u201d and \u201chas a chaise.\u201d For simplicity, let\u2019s use <em>S<\/em> to designate \u201cis a sectional,\u201d and <em>C<\/em> to designate \u201chas a chaise.\u201d The condition <em>S<\/em> is true if the couch is a sectional.\r\n\r\nA truth table for this would look like this:\r\n<table>\r\n<thead>\r\n<tr>\r\n<th><em>S<\/em><\/th>\r\n<th><em>C<\/em><\/th>\r\n<th><em>S<\/em> or\u00a0<em>C<\/em><\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the table, T is used for true, and F for false. In the first row, if <em>S<\/em> is true and <em>C<\/em> is also true, then the complex statement \u201c<em>S <\/em>or<em> C<\/em>\u201d is true. This would be a sectional that also has a chaise, which meets our desire.\r\n\r\nRemember also that <em>or<\/em> in logic is not exclusive; if the couch has both features, it does meet the condition.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nSome symbols that are commonly used for <em>and<\/em>, <em>or<\/em>, and <em>not<\/em> make using a truth table easier.\r\n<div class=\"textbox\">\r\n<h3>Symbols<\/h3>\r\nThe symbol [latex]\\wedge[\/latex] is used for <em>and<\/em>: <em>A<\/em> and <em>B<\/em> is notated [latex]A\\wedge{B}[\/latex].\r\n\r\nThe symbol [latex]\\vee[\/latex] is used for <em>or<\/em>: <em>A<\/em> or <em>B<\/em> is notated [latex]A\\vee{B}[\/latex]\r\n\r\nThe symbol [latex]\\sim[\/latex] is used for <em>not<\/em>: not <em>A<\/em> is notated [latex]\\sim{A}[\/latex]\r\n\r\n<\/div>\r\nYou can remember the first two symbols by relating them to the shapes for the union and intersection. [latex]A\\wedge{B}[\/latex]\u00a0would be the elements that exist in both sets, in [latex]A\\cap{B}[\/latex]. Likewise, [latex]A\\vee{B}[\/latex]\u00a0would be the elements that exist in either set, in [latex]A\\cup{B}[\/latex].\r\nIn the previous example, the truth table was really just summarizing what we already know about how the <em>or<\/em> statement work. The truth tables for the basic <em>and<\/em>, <em>or<\/em>, and <em>not<\/em> statements are shown below.\r\n<div class=\"textbox\">\r\n<h3>Basic Truth Tables<\/h3>\r\n<table width=\"40%&quot;\">\r\n<thead>\r\n<tr>\r\n<th>A<\/th>\r\n<th>B<\/th>\r\n<th>[latex]A\\wedge{B}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>A<\/th>\r\n<th>B<\/th>\r\n<th>[latex]A\\vee{B}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>A<\/th>\r\n<th>[latex]\\sim{A}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25467&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"150\"><\/iframe>\r\n\r\n<\/div>\r\nTruth tables really become useful when analyzing more complex Boolean statements.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nCreate a truth table for the statement [latex]A\\wedge\\sim\\left(B\\vee{C}\\right)[\/latex]\r\n[reveal-answer q=\"550652\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"550652\"]\r\n\r\nIt helps to work from the inside out when creating truth tables, and create tables for intermediate operations. We start by listing all the possible truth value combinations for <em>A<\/em>, <em>B<\/em>, and <em>C<\/em>. Notice how the first column contains 4 Ts followed by 4 Fs, the second column contains 2 Ts, 2 Fs, then repeats, and the last column alternates. This pattern ensures that all combinations are considered. Along with those initial values, we\u2019ll list the truth values for the innermost expression, [latex]B\\vee{C}[\/latex].\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>A<\/em><\/td>\r\n<td><em>B<\/em><\/td>\r\n<td><em>C<\/em><\/td>\r\n<td><em>B<\/em> \u22c1 <em>C<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNext we can find the negation of [latex]B\\vee{C}[\/latex], working off the [latex]B\\vee{C}[\/latex] column we just created.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>A<\/em><\/td>\r\n<td><em>B<\/em><\/td>\r\n<td><em>C<\/em><\/td>\r\n<td>[latex]B\\vee{C}[\/latex]<\/td>\r\n<td>[latex]\\sim\\left(B\\vee{C}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFinally, we find the values of <em>A<\/em> <em>and<\/em>\u00a0[latex]\\sim\\left(B\\vee{C}\\right)[\/latex]\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>A<\/em><\/td>\r\n<td><em>B<\/em><\/td>\r\n<td><em>C<\/em><\/td>\r\n<td>[latex]B\\vee{C}[\/latex]<\/td>\r\n<td>[latex]\\sim\\left(B\\vee{C}\\right)[\/latex]<\/td>\r\n<td>[latex]A\\wedge\\sim\\left(B{\\vee}C\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>\u00a0F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIt turns out that this complex expression is only true in one case: if A is true, B is false, and C is false.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25595&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"400\"><\/iframe>\r\n\r\n<\/div>\r\nWhen we discussed conditions earlier, we discussed the type where we take an action based on the value of the condition. We are now going to talk about a more general version of a conditional, sometimes called an <strong>implication<\/strong>.\r\n<div class=\"textbox\">\r\n<h3>Implications<\/h3>\r\nImplications are logical conditional sentences stating that a statement <em>p<\/em>, called the antecedent, implies a consequence <em>q<\/em>.\r\n\r\nImplications are commonly written as [latex]p\\rightarrow{q}[\/latex]\r\n\r\n<\/div>\r\nImplications are similar to the conditional statements we looked at earlier; [latex]p\\rightarrow{q}[\/latex] is typically written as \u201cif p then q,\u201d or \u201cp therefore q.\u201d The difference between implications and conditionals is that conditionals we discussed earlier suggest an action\u2014if the condition is true, then we take some action as a result. Implications are a logical statement that suggest that the consequence must logically follow if the antecedent is true.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe English statement \u201cIf it is raining, then there are clouds is the sky\u201d is a logical implication. Is this a valid argument, why or why not?\r\n[reveal-answer q=\"913754\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"913754\"]\r\n\r\nIt is a valid argument because if the antecedent \u201cit is raining\u201d is true, then the consequence \u201cthere are clouds in the sky\u201d must also be true.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nNotice that the statement tells us nothing of what to expect if it is not raining. If the antecedent is false, then the implication becomes irrelevant.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA friend tells you that \u201cif you upload that picture to Facebook, you\u2019ll lose your job.\u201d Describe the possible outcomes related to this statement, and determine whether your friend's statement is invalid.\r\n[reveal-answer q=\"463067\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"463067\"]\r\n\r\nThere are four possible outcomes:\r\n<ol>\r\n \t<li>You upload the picture and keep your job.<\/li>\r\n \t<li>You upload the picture and lose your job.<\/li>\r\n \t<li>You don\u2019t upload the picture and keep your job.<\/li>\r\n \t<li>You don\u2019t upload the picture and lose your job.<\/li>\r\n<\/ol>\r\nThere is only one possible case where your friend was lying\u2014the first option where you upload the picture and keep your job. In the last two cases, your friend didn\u2019t say anything about what would happen if you didn\u2019t upload the picture, so you can\u2019t conclude their statement is invalid, even if you didn\u2019t upload the picture and still lost your job.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn traditional logic, an implication is considered valid (true) as long as there are no cases in which the antecedent is true and the consequence is false. It is important to keep in mind that symbolic logic cannot capture all the intricacies of the English language.\r\n<div class=\"textbox\">\r\n<h3>Truth Values for Implications<\/h3>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>p<\/em><\/td>\r\n<td><em>q<\/em><\/td>\r\n<td><em>p<\/em> \u2192 <em>q<\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConstruct a truth table for the statement [latex]\\left(m\\wedge\\sim{p}\\right)\\rightarrow{r}[\/latex]\r\n\r\n[reveal-answer q=\"6001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"6001\"]\r\n\r\nWe start by constructing a truth table for the antecedent.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>m<\/em><\/td>\r\n<td><i>p<\/i><\/td>\r\n<td>[latex]\\sim{p}[\/latex]<\/td>\r\n<td>[latex]m\\wedge\\sim{p}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow we can build the truth table for the implication\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td><em>m<\/em><\/td>\r\n<td><i>p<\/i><\/td>\r\n<td>[latex]\\sim{p}[\/latex]<\/td>\r\n<td>[latex]m\\wedge\\sim{p}[\/latex]<\/td>\r\n<td><em>r<\/em><\/td>\r\n<td>[latex]\\left(m\\wedge\\sim{p}\\right)\\rightarrow{r}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn this case, when <em>m<\/em> is true, <em>p<\/em> is false, and <em>r<\/em> is false, then the antecedent [latex]m\\wedge\\sim{p}[\/latex]\u00a0will be true but the consequence false, resulting in a invalid implication; every other case gives a valid implication.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25597&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"400\"><\/iframe>\r\n\r\n<\/div>\r\nFor any implication, there are three related statements, the converse, the inverse, and the contrapositive.\r\n<div class=\"textbox\">\r\n<h3>Related Statements<\/h3>\r\nThe original implication is \u201cif <em>p<\/em> then <em>q<\/em>\u201d: [latex]p\\rightarrow{q}[\/latex]\r\n\r\nThe converse is \u201cif <em>q<\/em> then <em>p<\/em>\u201d: [latex]q\\rightarrow{p}[\/latex]\r\n\r\nThe inverse is \u201cif not <em>p<\/em> then not <em>q<\/em>\u201d: [latex]\\sim{p}\\rightarrow\\sim{q}[\/latex]\r\n\r\nThe contrapositive is \u201cif not <em>q<\/em> then not <em>p<\/em>\u201d: [latex]\\sim{q}\\rightarrow{p}[\/latex]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider again the valid implication \u201cIf it is raining, then there are clouds in the sky.\u201d\r\n\r\nWrite the related converse, inverse, and contrapositive statements.\r\n[reveal-answer q=\"746956\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"746956\"]\r\n\r\nThe converse would be \u201cIf there are clouds in the sky, it is raining.\u201d This is certainly not always true.\r\n\r\nThe inverse would be \u201cIf it is not raining, then there are not clouds in the sky.\u201d Likewise, this is not always true.\r\n\r\nThe contrapositive would be \u201cIf there are not clouds in the sky, then it is not raining.\u201d This statement is valid, and is equivalent to the original implication.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25472&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"400\"><\/iframe>\r\n\r\n<\/div>\r\nLooking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent, and that the converse and inverse are logically equivalent.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th><\/th>\r\n<th><\/th>\r\n<th>Implication<\/th>\r\n<th>Converse<\/th>\r\n<th>Inverse<\/th>\r\n<th>Contrapositive<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<th><em>p<\/em><\/th>\r\n<th><em>q<\/em><\/th>\r\n<th>[latex]p\\rightarrow{q}[\/latex]<\/th>\r\n<th>[latex]q{\\rightarrow}p[\/latex]<\/th>\r\n<th>[latex]\\sim{p}\\rightarrow\\sim{q}[\/latex]<\/th>\r\n<th>[latex]\\sim{q}\\rightarrow\\sim{p}[\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>F<\/td>\r\n<td>F<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<td>T<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox\">\r\n<h3>Equivalence<\/h3>\r\nA conditional statement and its contrapositive are logically equivalent.\r\n\r\nThe converse and inverse of a statement are logically equivalent.\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Combine sets using Boolean logic, using proper notations<\/li>\n<li>Use statements and conditionals to write and interpret expressions<\/li>\n<li>Use a truth table to interpret complex statements or conditionals<\/li>\n<li>Write truth tables given a logical implication, and it\u2019s related\u00a0statements \u2013 converse, inverse, and contrapositive<\/li>\n<li>Determine whether two statements are logically equivalent<\/li>\n<li>Use DeMorgan\u2019s laws to define logical equivalences of a statement<\/li>\n<\/ul>\n<\/div>\n<p>Because complex Boolean statements can get tricky to think about, we can create a <strong>truth table<\/strong> to break the complex statement into\u00a0simple statements, and determine whether they are true or false. A table will help keep track of all the truth values of the simple statements that make up a complex statement, leading to an analysis of the full statement.<\/p>\n<div class=\"textbox\">\n<h3>Truth Table<\/h3>\n<p>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<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Suppose you\u2019re picking out a new couch, and your significant other says \u201cget a sectional <em>or<\/em> something with a chaise.\u201d Construct a truth table that describes the elements of the conditions of this statement and whether the conditions are met.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q14714\">Show Solution<\/span><\/p>\n<div id=\"q14714\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is a complex statement made of two simpler conditions: \u201cis a sectional,\u201d and \u201chas a chaise.\u201d For simplicity, let\u2019s use <em>S<\/em> to designate \u201cis a sectional,\u201d and <em>C<\/em> to designate \u201chas a chaise.\u201d The condition <em>S<\/em> is true if the couch is a sectional.<\/p>\n<p>A truth table for this would look like this:<\/p>\n<table>\n<thead>\n<tr>\n<th><em>S<\/em><\/th>\n<th><em>C<\/em><\/th>\n<th><em>S<\/em> or\u00a0<em>C<\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the table, T is used for true, and F for false. In the first row, if <em>S<\/em> is true and <em>C<\/em> is also true, then the complex statement \u201c<em>S <\/em>or<em> C<\/em>\u201d is true. This would be a sectional that also has a chaise, which meets our desire.<\/p>\n<p>Remember also that <em>or<\/em> in logic is not exclusive; if the couch has both features, it does meet the condition.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Some symbols that are commonly used for <em>and<\/em>, <em>or<\/em>, and <em>not<\/em> make using a truth table easier.<\/p>\n<div class=\"textbox\">\n<h3>Symbols<\/h3>\n<p>The symbol [latex]\\wedge[\/latex] is used for <em>and<\/em>: <em>A<\/em> and <em>B<\/em> is notated [latex]A\\wedge{B}[\/latex].<\/p>\n<p>The symbol [latex]\\vee[\/latex] is used for <em>or<\/em>: <em>A<\/em> or <em>B<\/em> is notated [latex]A\\vee{B}[\/latex]<\/p>\n<p>The symbol [latex]\\sim[\/latex] is used for <em>not<\/em>: not <em>A<\/em> is notated [latex]\\sim{A}[\/latex]<\/p>\n<\/div>\n<p>You can remember the first two symbols by relating them to the shapes for the union and intersection. [latex]A\\wedge{B}[\/latex]\u00a0would be the elements that exist in both sets, in [latex]A\\cap{B}[\/latex]. Likewise, [latex]A\\vee{B}[\/latex]\u00a0would be the elements that exist in either set, in [latex]A\\cup{B}[\/latex].<br \/>\nIn the previous example, the truth table was really just summarizing what we already know about how the <em>or<\/em> statement work. The truth tables for the basic <em>and<\/em>, <em>or<\/em>, and <em>not<\/em> statements are shown below.<\/p>\n<div class=\"textbox\">\n<h3>Basic Truth Tables<\/h3>\n<table style=\"width: 40%&quot;\">\n<thead>\n<tr>\n<th>A<\/th>\n<th>B<\/th>\n<th>[latex]A\\wedge{B}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<thead>\n<tr>\n<th>A<\/th>\n<th>B<\/th>\n<th>[latex]A\\vee{B}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<thead>\n<tr>\n<th>A<\/th>\n<th>[latex]\\sim{A}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25467&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Truth tables really become useful when analyzing more complex Boolean statements.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Create a truth table for the statement [latex]A\\wedge\\sim\\left(B\\vee{C}\\right)[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q550652\">Show Solution<\/span><\/p>\n<div id=\"q550652\" class=\"hidden-answer\" style=\"display: none\">\n<p>It helps to work from the inside out when creating truth tables, and create tables for intermediate operations. We start by listing all the possible truth value combinations for <em>A<\/em>, <em>B<\/em>, and <em>C<\/em>. Notice how the first column contains 4 Ts followed by 4 Fs, the second column contains 2 Ts, 2 Fs, then repeats, and the last column alternates. This pattern ensures that all combinations are considered. Along with those initial values, we\u2019ll list the truth values for the innermost expression, [latex]B\\vee{C}[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>A<\/em><\/td>\n<td><em>B<\/em><\/td>\n<td><em>C<\/em><\/td>\n<td><em>B<\/em> \u22c1 <em>C<\/em><\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Next we can find the negation of [latex]B\\vee{C}[\/latex], working off the [latex]B\\vee{C}[\/latex] column we just created.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>A<\/em><\/td>\n<td><em>B<\/em><\/td>\n<td><em>C<\/em><\/td>\n<td>[latex]B\\vee{C}[\/latex]<\/td>\n<td>[latex]\\sim\\left(B\\vee{C}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Finally, we find the values of <em>A<\/em> <em>and<\/em>\u00a0[latex]\\sim\\left(B\\vee{C}\\right)[\/latex]<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>A<\/em><\/td>\n<td><em>B<\/em><\/td>\n<td><em>C<\/em><\/td>\n<td>[latex]B\\vee{C}[\/latex]<\/td>\n<td>[latex]\\sim\\left(B\\vee{C}\\right)[\/latex]<\/td>\n<td>[latex]A\\wedge\\sim\\left(B{\\vee}C\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>\u00a0F<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>It turns out that this complex expression is only true in one case: if A is true, B is false, and C is false.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25595&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n<p>When we discussed conditions earlier, we discussed the type where we take an action based on the value of the condition. We are now going to talk about a more general version of a conditional, sometimes called an <strong>implication<\/strong>.<\/p>\n<div class=\"textbox\">\n<h3>Implications<\/h3>\n<p>Implications are logical conditional sentences stating that a statement <em>p<\/em>, called the antecedent, implies a consequence <em>q<\/em>.<\/p>\n<p>Implications are commonly written as [latex]p\\rightarrow{q}[\/latex]<\/p>\n<\/div>\n<p>Implications are similar to the conditional statements we looked at earlier; [latex]p\\rightarrow{q}[\/latex] is typically written as \u201cif p then q,\u201d or \u201cp therefore q.\u201d The difference between implications and conditionals is that conditionals we discussed earlier suggest an action\u2014if the condition is true, then we take some action as a result. Implications are a logical statement that suggest that the consequence must logically follow if the antecedent is true.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The English statement \u201cIf it is raining, then there are clouds is the sky\u201d is a logical implication. Is this a valid argument, why or why not?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q913754\">Show Solution<\/span><\/p>\n<div id=\"q913754\" class=\"hidden-answer\" style=\"display: none\">\n<p>It is a valid argument because if the antecedent \u201cit is raining\u201d is true, then the consequence \u201cthere are clouds in the sky\u201d must also be true.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Notice that the statement tells us nothing of what to expect if it is not raining. If the antecedent is false, then the implication becomes irrelevant.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A friend tells you that \u201cif you upload that picture to Facebook, you\u2019ll lose your job.\u201d Describe the possible outcomes related to this statement, and determine whether your friend&#8217;s statement is invalid.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q463067\">Show Solution<\/span><\/p>\n<div id=\"q463067\" class=\"hidden-answer\" style=\"display: none\">\n<p>There are four possible outcomes:<\/p>\n<ol>\n<li>You upload the picture and keep your job.<\/li>\n<li>You upload the picture and lose your job.<\/li>\n<li>You don\u2019t upload the picture and keep your job.<\/li>\n<li>You don\u2019t upload the picture and lose your job.<\/li>\n<\/ol>\n<p>There is only one possible case where your friend was lying\u2014the first option where you upload the picture and keep your job. In the last two cases, your friend didn\u2019t say anything about what would happen if you didn\u2019t upload the picture, so you can\u2019t conclude their statement is invalid, even if you didn\u2019t upload the picture and still lost your job.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In traditional logic, an implication is considered valid (true) as long as there are no cases in which the antecedent is true and the consequence is false. It is important to keep in mind that symbolic logic cannot capture all the intricacies of the English language.<\/p>\n<div class=\"textbox\">\n<h3>Truth Values for Implications<\/h3>\n<table>\n<tbody>\n<tr>\n<td><em>p<\/em><\/td>\n<td><em>q<\/em><\/td>\n<td><em>p<\/em> \u2192 <em>q<\/em><\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Construct a truth table for the statement [latex]\\left(m\\wedge\\sim{p}\\right)\\rightarrow{r}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q6001\">Show Solution<\/span><\/p>\n<div id=\"q6001\" class=\"hidden-answer\" style=\"display: none\">\n<p>We start by constructing a truth table for the antecedent.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>m<\/em><\/td>\n<td><i>p<\/i><\/td>\n<td>[latex]\\sim{p}[\/latex]<\/td>\n<td>[latex]m\\wedge\\sim{p}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now we can build the truth table for the implication<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>m<\/em><\/td>\n<td><i>p<\/i><\/td>\n<td>[latex]\\sim{p}[\/latex]<\/td>\n<td>[latex]m\\wedge\\sim{p}[\/latex]<\/td>\n<td><em>r<\/em><\/td>\n<td>[latex]\\left(m\\wedge\\sim{p}\\right)\\rightarrow{r}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In this case, when <em>m<\/em> is true, <em>p<\/em> is false, and <em>r<\/em> is false, then the antecedent [latex]m\\wedge\\sim{p}[\/latex]\u00a0will be true but the consequence false, resulting in a invalid implication; every other case gives a valid implication.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25597&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n<p>For any implication, there are three related statements, the converse, the inverse, and the contrapositive.<\/p>\n<div class=\"textbox\">\n<h3>Related Statements<\/h3>\n<p>The original implication is \u201cif <em>p<\/em> then <em>q<\/em>\u201d: [latex]p\\rightarrow{q}[\/latex]<\/p>\n<p>The converse is \u201cif <em>q<\/em> then <em>p<\/em>\u201d: [latex]q\\rightarrow{p}[\/latex]<\/p>\n<p>The inverse is \u201cif not <em>p<\/em> then not <em>q<\/em>\u201d: [latex]\\sim{p}\\rightarrow\\sim{q}[\/latex]<\/p>\n<p>The contrapositive is \u201cif not <em>q<\/em> then not <em>p<\/em>\u201d: [latex]\\sim{q}\\rightarrow{p}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider again the valid implication \u201cIf it is raining, then there are clouds in the sky.\u201d<\/p>\n<p>Write the related converse, inverse, and contrapositive statements.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q746956\">Show Solution<\/span><\/p>\n<div id=\"q746956\" class=\"hidden-answer\" style=\"display: none\">\n<p>The converse would be \u201cIf there are clouds in the sky, it is raining.\u201d This is certainly not always true.<\/p>\n<p>The inverse would be \u201cIf it is not raining, then there are not clouds in the sky.\u201d Likewise, this is not always true.<\/p>\n<p>The contrapositive would be \u201cIf there are not clouds in the sky, then it is not raining.\u201d This statement is valid, and is equivalent to the original implication.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=25472&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"400\"><\/iframe><\/p>\n<\/div>\n<p>Looking at truth tables, we can see that the original conditional and the contrapositive are logically equivalent, and that the converse and inverse are logically equivalent.<\/p>\n<table>\n<thead>\n<tr>\n<th><\/th>\n<th><\/th>\n<th>Implication<\/th>\n<th>Converse<\/th>\n<th>Inverse<\/th>\n<th>Contrapositive<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th><em>p<\/em><\/th>\n<th><em>q<\/em><\/th>\n<th>[latex]p\\rightarrow{q}[\/latex]<\/th>\n<th>[latex]q{\\rightarrow}p[\/latex]<\/th>\n<th>[latex]\\sim{p}\\rightarrow\\sim{q}[\/latex]<\/th>\n<th>[latex]\\sim{q}\\rightarrow\\sim{p}[\/latex]<\/th>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox\">\n<h3>Equivalence<\/h3>\n<p>A conditional statement and its contrapositive are logically equivalent.<\/p>\n<p>The converse and inverse of a statement are logically equivalent.<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-864\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Question ID 25472, 25467. <strong>Authored by<\/strong>: Shahbazian,Roy. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 25595, 25597. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Question ID 25472, 25467\",\"author\":\"Shahbazian,Roy\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\"Question ID 25595, 25597\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"3272ce56-7979-4f7a-8baf-acae8ee560cb","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-864","chapter","type-chapter","status-publish","hentry"],"part":159,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/864","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/users\/21"}],"version-history":[{"count":16,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/864\/revisions"}],"predecessor-version":[{"id":2985,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/864\/revisions\/2985"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/parts\/159"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapters\/864\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/media?parent=864"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/pressbooks\/v2\/chapter-type?post=864"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/contributor?post=864"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/waymakermath4libarts\/wp-json\/wp\/v2\/license?post=864"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}