{"id":203,"date":"2023-06-05T15:30:14","date_gmt":"2023-06-05T15:30:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/chapter\/truth-tables\/"},"modified":"2024-03-12T20:27:11","modified_gmt":"2024-03-12T20:27:11","slug":"truth-tables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/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>Use a truth table to interpret complex statements or conditionals<\/li>\r\n \t<li>Write truth tables given a logical implication, and its related\u00a0statements<\/li>\r\n \t<li>Determine whether two statements are logically equivalent<\/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 examples\">\r\n<h3>study strategy<\/h3>\r\nYou may notice that you've accumulated quite a bit of new vocabulary and symbols. A helpful technique is to collect all of these in a central location: a set of flashcards, a notebook, or something similar.\r\n\r\nNew notation and vocabulary are introduced in this page as well. Try to find similarities between the symbols in this page and the ones you encountered in previous pages in this module.\r\n\r\n<\/div>\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.\u00a0 It is a good idea to try to memorize these basic truth tables.\u00a0 It will help you when you are working with truth tables and more complex statements.\r\n<div class=\"textbox\">\r\n<h3>Basic Truth Tables<\/h3>\r\n<strong>\"And\" - Conjunction:\u00a0 A\u00a0\u039b B<\/strong>\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<strong>\"Or\" - Disjunction: A V B<\/strong>\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<strong>\"Not: - Negation: ~A<\/strong>\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<a href=\"https:\/\/youtu.be\/_5Z_0824RHw\" target=\"_blank\" rel=\"noopener\">Click Here<\/a> to see a video going over truth tables for conjunction and disjunction.\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]25467[\/ohm_question]\r\n\r\n<\/div>\r\nTruth tables really become useful when analyzing more complex Boolean statements.\u00a0 But, what is the correct order needed when completing a truth table?\r\n<h2 data-type=\"title\">The Dominance of Connectives<\/h2>\r\n<p id=\"para-00040\">The order of operations for working with algebraic and arithmetic expressions provides a set of rules that allow consistent results. For example, if you were presented with the problem <span class=\"os-math-in-para\"><span id=\"MathJax-Element-741-Frame\" class=\"MathJax\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;&amp;#xD7;&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;\u00d7&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-6004\" class=\"math\"><span id=\"MathJax-Span-6005\" class=\"mrow\"><span id=\"MathJax-Span-6006\" class=\"semantics\"><span id=\"MathJax-Span-6007\" class=\"mrow\"><span id=\"MathJax-Span-6008\" class=\"mrow\"><span id=\"MathJax-Span-6009\" class=\"mn\">1<\/span><span id=\"MathJax-Span-6010\" class=\"mo\">+<\/span><span id=\"MathJax-Span-6011\" class=\"mn\">3<\/span><span id=\"MathJax-Span-6012\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-6013\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, and you were not familiar with the order of operation, you might assume that you calculate the problem from left to right. If you did so, you would add 1 and 3 to get 4, and then multiply this answer by 2 to get 8, resulting in an incorrect answer. Try inputting this expression into a scientific calculator. If you do, the calculator should return a value of 7, not 8.<\/p>\r\n<p id=\"para-00041\"><a href=\"https:\/\/openstax.org\/r\/Scientific_Calculator\" target=\"_blank\" rel=\"noopener nofollow\">Scientific Calculator<\/a><\/p>\r\n<p id=\"para-00042\">The order of operations for algebraic and arithmetic operations states that all multiplication must be applied prior to any addition. Parentheses are used to indicate which operation\u2014addition or multiplication\u2014should be done first. Adding parentheses can change and\/or clarify the order. The parentheses in the expression <span class=\"os-math-in-para\"><span id=\"MathJax-Element-742-Frame\" class=\"MathJax\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;\/mo&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;&amp;#xD7;&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;mo&gt;)&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;\/mo&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;\u00d7&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;mo&gt;)&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-6014\" class=\"math\"><span id=\"MathJax-Span-6015\" class=\"mrow\"><span id=\"MathJax-Span-6016\" class=\"semantics\"><span id=\"MathJax-Span-6017\" class=\"mrow\"><span id=\"MathJax-Span-6018\" class=\"mrow\"><span id=\"MathJax-Span-6019\" class=\"mn\">1<\/span><span id=\"MathJax-Span-6020\" class=\"mo\">+<\/span><span id=\"MathJax-Span-6021\" class=\"mrow\"><span id=\"MathJax-Span-6022\" class=\"mo\">(<\/span><span id=\"MathJax-Span-6023\" class=\"mrow\"><span id=\"MathJax-Span-6024\" class=\"mn\">3<\/span><span id=\"MathJax-Span-6025\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-6026\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-6027\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> indicate that 3 should be multiplied by 2 to get 6, and then 1 should be added to 6 to get 7: <span class=\"os-math-in-para\"><span id=\"MathJax-Element-743-Frame\" class=\"MathJax\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;\/mo&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;&amp;#xD7;&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;mo&gt;)&lt;\/mo&gt;&lt;\/mrow&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mn&gt;7.&lt;\/mn&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;\/mo&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;\u00d7&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;mo&gt;)&lt;\/mo&gt;&lt;\/mrow&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mn&gt;7.&lt;\/mn&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-6028\" class=\"math\"><span id=\"MathJax-Span-6029\" class=\"mrow\"><span id=\"MathJax-Span-6030\" class=\"semantics\"><span id=\"MathJax-Span-6031\" class=\"mrow\"><span id=\"MathJax-Span-6032\" class=\"mrow\"><span id=\"MathJax-Span-6033\" class=\"mn\">1<\/span><span id=\"MathJax-Span-6034\" class=\"mo\">+<\/span><span id=\"MathJax-Span-6035\" class=\"mrow\"><span id=\"MathJax-Span-6036\" class=\"mo\">(<\/span><span id=\"MathJax-Span-6037\" class=\"mrow\"><span id=\"MathJax-Span-6038\" class=\"mn\">3<\/span><span id=\"MathJax-Span-6039\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-6040\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-6041\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-6042\" class=\"mo\">=<\/span><span id=\"MathJax-Span-6043\" class=\"mn\">7.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\r\n<p id=\"para-00043\">As with algebraic expressions, there is a set of rules that must be applied to compound logical statements in order to evaluate them with consistent results. This set of rules is called the <span id=\"term-00008\" data-type=\"term\">dominance of connectives<\/span>. When evaluating compound logical statements, connectives are evaluated from least dominant to most dominant as follows:<\/p>\r\n\r\n<ol id=\"list-00012\" type=\"1\">\r\n \t<li>Parentheses are the least dominant connective. So, any expression inside parentheses must be evaluated first. Add as many parentheses as needed to any statement to specify the order to evaluate each connective.<\/li>\r\n \t<li>Next, we evaluate negations.<\/li>\r\n \t<li>Then, we evaluate conjunctions and disjunctions from left to right, because they have equal dominance.<\/li>\r\n \t<li>After evaluating all conjunctions and disjunctions, we evaluate conditionals.<\/li>\r\n \t<li>Lastly, we evaluate the most dominant connective, the biconditional. If a statement includes multiple connectives of equal dominance, then we will evaluate them from left to right.<\/li>\r\n<\/ol>\r\n<p id=\"para-00005\">The following image is a visual breakdown of the dominance of connectives.<\/p>\r\n\r\n<div id=\"fig-00003\" class=\"os-figure\">\r\n<figure data-id=\"fig-00003\"><span data-type=\"media\" data-alt=\"A table with four columns shows Dominance, Connective, Symbol, and Evaluate. The dominance column on the table shows a downward vertical arrow from least dominant to most dominant. The connective column on the table shows Parentheses, Negation, Disjunction or Conjunction, Conditional, and Biconditional. The Symbol column on the table shows an open bracket and a closed bracket, equivalent, an upward circumflex and a downward circumflex, a right side arrow, and a double-sided arrow. The Evaluate column on the table shows First, a downward arrow, Left to right or add parentheses to specify order because or slash and have equal dominance. a downward arrow, and last.\"> <img src=\"https:\/\/openstax.org\/apps\/archive\/20240130.204924\/resources\/11dc692a57f932d834486dcbaeed7c5082cf662f\" alt=\"A table with four columns shows Dominance, Connective, Symbol, and Evaluate. The dominance column on the table shows a downward vertical arrow from least dominant to most dominant. The connective column on the table shows Parentheses, Negation, Disjunction or Conjunction, Conditional, and Biconditional. The Symbol column on the table shows an open bracket and a closed bracket, equivalent, an upward circumflex and a downward circumflex, a right side arrow, and a double-sided arrow. The Evaluate column on the table shows First, a downward arrow, Left to right or add parentheses to specify order because or slash and have equal dominance. a downward arrow, and last.\" width=\"1147\" height=\"414\" data-media-type=\"image\/jpg\" \/> <\/span><\/figure>\r\n<\/div>\r\n&nbsp;\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[ohm_question]25595[\/ohm_question]\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 (conditionals) and Biconditionals<\/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<a href=\"https:\/\/youtu.be\/2npo-L0DJRQ\">Click Here<\/a> to see a short video over truth tables for conditionals.\r\n\r\nThere is another way to look at conditionals - that is, if P then Q and If Q then P.\u00a0 We call these \"biconditionals\" and we write them as \"P if and only if Q\".\u00a0 Symbolically, we write this as P\u00a0\u2194 Q.\u00a0 Below is the truth table for this biconditional.\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 \u2194<\/em>\u00a0<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>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<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nA biconditional is considered to be true if both P and Q are true simultaneously or if both P and Q are false simultaneously.\r\n\r\n<a href=\"https:\/\/youtu.be\/-r8FzV84sj8\">Click Here<\/a> to see a short video over the truth table for a biconditional.\r\n\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[ohm_question]25597[\/ohm_question]\r\n\r\n<\/div>\r\nFor any implication, there are three related statements, the <strong>converse<\/strong>, the <strong>inverse<\/strong>, and the <strong>contrapositive<\/strong>.\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\\sim{p}[\/latex]\r\n\r\n<a href=\"https:\/\/youtu.be\/IHd8jiUF3Lk\">Click here<\/a> to see a video for the truth tables for each of the related statements.\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[ohm_question]25472[\/ohm_question]\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>Use a truth table to interpret complex statements or conditionals<\/li>\n<li>Write truth tables given a logical implication, and its related\u00a0statements<\/li>\n<li>Determine whether two statements are logically equivalent<\/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 examples\">\n<h3>study strategy<\/h3>\n<p>You may notice that you&#8217;ve accumulated quite a bit of new vocabulary and symbols. A helpful technique is to collect all of these in a central location: a set of flashcards, a notebook, or something similar.<\/p>\n<p>New notation and vocabulary are introduced in this page as well. Try to find similarities between the symbols in this page and the ones you encountered in previous pages in this module.<\/p>\n<\/div>\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.\u00a0 It is a good idea to try to memorize these basic truth tables.\u00a0 It will help you when you are working with truth tables and more complex statements.<\/p>\n<div class=\"textbox\">\n<h3>Basic Truth Tables<\/h3>\n<p><strong>&#8220;And&#8221; &#8211; Conjunction:\u00a0 A\u00a0\u039b B<\/strong><\/p>\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<p><strong>&#8220;Or&#8221; &#8211; Disjunction: A V B<\/strong><\/p>\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<p><strong>&#8220;Not: &#8211; Negation: ~A<\/strong><\/p>\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<p><a href=\"https:\/\/youtu.be\/_5Z_0824RHw\" target=\"_blank\" rel=\"noopener\">Click Here<\/a> to see a video going over truth tables for conjunction and disjunction.<\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm25467\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25467&theme=oea&iframe_resize_id=ohm25467&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>Truth tables really become useful when analyzing more complex Boolean statements.\u00a0 But, what is the correct order needed when completing a truth table?<\/p>\n<h2 data-type=\"title\">The Dominance of Connectives<\/h2>\n<p id=\"para-00040\">The order of operations for working with algebraic and arithmetic expressions provides a set of rules that allow consistent results. For example, if you were presented with the problem <span class=\"os-math-in-para\"><span id=\"MathJax-Element-741-Frame\" class=\"MathJax\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;&amp;#xD7;&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;\u00d7&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-6004\" class=\"math\"><span id=\"MathJax-Span-6005\" class=\"mrow\"><span id=\"MathJax-Span-6006\" class=\"semantics\"><span id=\"MathJax-Span-6007\" class=\"mrow\"><span id=\"MathJax-Span-6008\" class=\"mrow\"><span id=\"MathJax-Span-6009\" class=\"mn\">1<\/span><span id=\"MathJax-Span-6010\" class=\"mo\">+<\/span><span id=\"MathJax-Span-6011\" class=\"mn\">3<\/span><span id=\"MathJax-Span-6012\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-6013\" class=\"mn\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, and you were not familiar with the order of operation, you might assume that you calculate the problem from left to right. If you did so, you would add 1 and 3 to get 4, and then multiply this answer by 2 to get 8, resulting in an incorrect answer. Try inputting this expression into a scientific calculator. If you do, the calculator should return a value of 7, not 8.<\/p>\n<p id=\"para-00041\"><a href=\"https:\/\/openstax.org\/r\/Scientific_Calculator\" target=\"_blank\" rel=\"noopener nofollow\">Scientific Calculator<\/a><\/p>\n<p id=\"para-00042\">The order of operations for algebraic and arithmetic operations states that all multiplication must be applied prior to any addition. Parentheses are used to indicate which operation\u2014addition or multiplication\u2014should be done first. Adding parentheses can change and\/or clarify the order. The parentheses in the expression <span class=\"os-math-in-para\"><span id=\"MathJax-Element-742-Frame\" class=\"MathJax\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;\/mo&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;&amp;#xD7;&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;mo&gt;)&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;\/mo&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;\u00d7&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;mo&gt;)&lt;\/mo&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-6014\" class=\"math\"><span id=\"MathJax-Span-6015\" class=\"mrow\"><span id=\"MathJax-Span-6016\" class=\"semantics\"><span id=\"MathJax-Span-6017\" class=\"mrow\"><span id=\"MathJax-Span-6018\" class=\"mrow\"><span id=\"MathJax-Span-6019\" class=\"mn\">1<\/span><span id=\"MathJax-Span-6020\" class=\"mo\">+<\/span><span id=\"MathJax-Span-6021\" class=\"mrow\"><span id=\"MathJax-Span-6022\" class=\"mo\">(<\/span><span id=\"MathJax-Span-6023\" class=\"mrow\"><span id=\"MathJax-Span-6024\" class=\"mn\">3<\/span><span id=\"MathJax-Span-6025\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-6026\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-6027\" class=\"mo\">)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> indicate that 3 should be multiplied by 2 to get 6, and then 1 should be added to 6 to get 7: <span class=\"os-math-in-para\"><span id=\"MathJax-Element-743-Frame\" class=\"MathJax\" tabindex=\"0\" role=\"presentation\" data-mathml=\"&lt;math xmlns=&quot;http:\/\/www.w3.org\/1998\/Math\/MathML&quot; display=&quot;inline&quot;&gt;&lt;semantics&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;\/mo&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;&amp;#xD7;&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;mo&gt;)&lt;\/mo&gt;&lt;\/mrow&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mn&gt;7.&lt;\/mn&gt;&lt;\/mrow&gt;&lt;\/mrow&gt;&lt;annotation-xml encoding=&quot;MathML-Content&quot;&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;\/mn&gt;&lt;mo&gt;+&lt;\/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;\/mo&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;\/mn&gt;&lt;mo&gt;\u00d7&lt;\/mo&gt;&lt;mn&gt;2&lt;\/mn&gt;&lt;\/mrow&gt;&lt;mo&gt;)&lt;\/mo&gt;&lt;\/mrow&gt;&lt;mo&gt;=&lt;\/mo&gt;&lt;mn&gt;7.&lt;\/mn&gt;&lt;\/mrow&gt;&lt;\/annotation-xml&gt;&lt;\/semantics&gt;&lt;\/math&gt;\"><span id=\"MathJax-Span-6028\" class=\"math\"><span id=\"MathJax-Span-6029\" class=\"mrow\"><span id=\"MathJax-Span-6030\" class=\"semantics\"><span id=\"MathJax-Span-6031\" class=\"mrow\"><span id=\"MathJax-Span-6032\" class=\"mrow\"><span id=\"MathJax-Span-6033\" class=\"mn\">1<\/span><span id=\"MathJax-Span-6034\" class=\"mo\">+<\/span><span id=\"MathJax-Span-6035\" class=\"mrow\"><span id=\"MathJax-Span-6036\" class=\"mo\">(<\/span><span id=\"MathJax-Span-6037\" class=\"mrow\"><span id=\"MathJax-Span-6038\" class=\"mn\">3<\/span><span id=\"MathJax-Span-6039\" class=\"mo\">\u00d7<\/span><span id=\"MathJax-Span-6040\" class=\"mn\">2<\/span><\/span><span id=\"MathJax-Span-6041\" class=\"mo\">)<\/span><\/span><span id=\"MathJax-Span-6042\" class=\"mo\">=<\/span><span id=\"MathJax-Span-6043\" class=\"mn\">7.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p id=\"para-00043\">As with algebraic expressions, there is a set of rules that must be applied to compound logical statements in order to evaluate them with consistent results. This set of rules is called the <span id=\"term-00008\" data-type=\"term\">dominance of connectives<\/span>. When evaluating compound logical statements, connectives are evaluated from least dominant to most dominant as follows:<\/p>\n<ol id=\"list-00012\" type=\"1\">\n<li>Parentheses are the least dominant connective. So, any expression inside parentheses must be evaluated first. Add as many parentheses as needed to any statement to specify the order to evaluate each connective.<\/li>\n<li>Next, we evaluate negations.<\/li>\n<li>Then, we evaluate conjunctions and disjunctions from left to right, because they have equal dominance.<\/li>\n<li>After evaluating all conjunctions and disjunctions, we evaluate conditionals.<\/li>\n<li>Lastly, we evaluate the most dominant connective, the biconditional. If a statement includes multiple connectives of equal dominance, then we will evaluate them from left to right.<\/li>\n<\/ol>\n<p id=\"para-00005\">The following image is a visual breakdown of the dominance of connectives.<\/p>\n<div id=\"fig-00003\" class=\"os-figure\">\n<figure data-id=\"fig-00003\"><span data-type=\"media\" data-alt=\"A table with four columns shows Dominance, Connective, Symbol, and Evaluate. The dominance column on the table shows a downward vertical arrow from least dominant to most dominant. The connective column on the table shows Parentheses, Negation, Disjunction or Conjunction, Conditional, and Biconditional. The Symbol column on the table shows an open bracket and a closed bracket, equivalent, an upward circumflex and a downward circumflex, a right side arrow, and a double-sided arrow. The Evaluate column on the table shows First, a downward arrow, Left to right or add parentheses to specify order because or slash and have equal dominance. a downward arrow, and last.\"> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/apps\/archive\/20240130.204924\/resources\/11dc692a57f932d834486dcbaeed7c5082cf662f\" alt=\"A table with four columns shows Dominance, Connective, Symbol, and Evaluate. The dominance column on the table shows a downward vertical arrow from least dominant to most dominant. The connective column on the table shows Parentheses, Negation, Disjunction or Conjunction, Conditional, and Biconditional. The Symbol column on the table shows an open bracket and a closed bracket, equivalent, an upward circumflex and a downward circumflex, a right side arrow, and a double-sided arrow. The Evaluate column on the table shows First, a downward arrow, Left to right or add parentheses to specify order because or slash and have equal dominance. a downward arrow, and last.\" width=\"1147\" height=\"414\" data-media-type=\"image\/jpg\" \/> <\/span><\/figure>\n<\/div>\n<p>&nbsp;<\/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=\"ohm25595\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25595&theme=oea&iframe_resize_id=ohm25595&show_question_numbers\" width=\"100%\" height=\"150\"><\/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 (conditionals) and Biconditionals<\/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<p><a href=\"https:\/\/youtu.be\/2npo-L0DJRQ\">Click Here<\/a> to see a short video over truth tables for conditionals.<\/p>\n<p>There is another way to look at conditionals &#8211; that is, if P then Q and If Q then P.\u00a0 We call these &#8220;biconditionals&#8221; and we write them as &#8220;P if and only if Q&#8221;.\u00a0 Symbolically, we write this as P\u00a0\u2194 Q.\u00a0 Below is the truth table for this biconditional.<\/p>\n<table>\n<tbody>\n<tr>\n<td><em>p<\/em><\/td>\n<td><em>q<\/em><\/td>\n<td><em>p \u2194<\/em>\u00a0<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>F<\/td>\n<\/tr>\n<tr>\n<td>F<\/td>\n<td>F<\/td>\n<td>T<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>A biconditional is considered to be true if both P and Q are true simultaneously or if both P and Q are false simultaneously.<\/p>\n<p><a href=\"https:\/\/youtu.be\/-r8FzV84sj8\">Click Here<\/a> to see a short video over the truth table for a biconditional.<\/p>\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=\"ohm25597\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25597&theme=oea&iframe_resize_id=ohm25597&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>For any implication, there are three related statements, the <strong>converse<\/strong>, the <strong>inverse<\/strong>, and the <strong>contrapositive<\/strong>.<\/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\\sim{p}[\/latex]<\/p>\n<p><a href=\"https:\/\/youtu.be\/IHd8jiUF3Lk\">Click here<\/a> to see a video for the truth tables for each of the related statements.<\/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=\"ohm25472\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25472&theme=oea&iframe_resize_id=ohm25472&show_question_numbers\" width=\"100%\" height=\"150\"><\/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-203\">\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":503070,"menu_order":10,"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-203","chapter","type-chapter","status-publish","hentry"],"part":184,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/pressbooks\/v2\/chapters\/203","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/wp\/v2\/users\/503070"}],"version-history":[{"count":6,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/pressbooks\/v2\/chapters\/203\/revisions"}],"predecessor-version":[{"id":1185,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/pressbooks\/v2\/chapters\/203\/revisions\/1185"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/pressbooks\/v2\/parts\/184"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/pressbooks\/v2\/chapters\/203\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/wp\/v2\/media?parent=203"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/pressbooks\/v2\/chapter-type?post=203"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/wp\/v2\/contributor?post=203"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1473\/wp-json\/wp\/v2\/license?post=203"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}