{"id":6793,"date":"2021-12-30T21:43:26","date_gmt":"2021-12-30T21:43:26","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=6793"},"modified":"2022-01-06T22:23:27","modified_gmt":"2022-01-06T22:23:27","slug":"1-10-interpreting-truth-tables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/1-10-interpreting-truth-tables\/","title":{"raw":"1.10 Interpreting Truth Tables","rendered":"1.10 Interpreting Truth Tables"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nInterpreting Truth Tables\r\n<ul>\r\n \t<li>Tautology<\/li>\r\n \t<li>Contradiction<\/li>\r\n \t<li>Contingency<\/li>\r\n \t<li>Logically Equivalent<\/li>\r\n<\/ul>\r\n<\/div>\r\nA <strong>tautology<\/strong> is a proposition that is always true, regardless of the truth values of the propositional variables it contains. A proposition that is always false is called a <strong>contradiction<\/strong>. A proposition that is neither a tautology nor a contradiction is called a <strong>contingency<\/strong>.\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nAn example of a <strong>tautology<\/strong> is:\u00a0 I am going to take Math for Liberal Arts this semester or I' m not going to take Math for Liberal Arts this semester.\u00a0 This statement is always true so it is a tautology.\r\n\r\nAn example of a <strong>self-contradition<\/strong> is:\u00a0 I will get an A in this class and I will not get an A in this class.\u00a0 This statement is always false so it is a self-contradiction.\r\n\r\n&nbsp;\r\n\r\nCaution:\u00a0 Don't make the mistake that every statement is either a tautology or a self-contradiction.\u00a0 We have seen many examples of truth tables which have a mixture of trues and falses in the final column.\u00a0 These statements are sometimes true and sometimes not true.\u00a0 These are called a <strong>contingency<\/strong>.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nComplete a truth table to classify the statement as tautology, self-contradiction or contingency.\r\n\r\na.\u00a0 p\u00a0<img class=\"\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>\u00a0(~p)\r\n\r\nb.\u00a0 (p\u00a0<img class=\"\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" width=\"8\" height=\"8\" \/>\u00a0q)\u00a0<img class=\"\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" width=\"8\" height=\"8\" \/>\u00a0(~p\u00a0<img class=\"\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>\u00a0~q)\r\n\r\nSolution:\r\n\r\na.\u00a0 This is a tautology\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/03024926\/tautology.png\"><img class=\"alignnone wp-image-6900\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/03024926\/tautology.png\" alt=\"\" width=\"164\" height=\"119\" \/><\/a>\r\n\r\nb.\u00a0 This is a self-contradiction\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/03025325\/self-contradiction.png\"><img class=\"alignnone wp-image-6901\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/03025325\/self-contradiction-300x89.png\" alt=\"\" width=\"418\" height=\"124\" \/><\/a>\r\n\r\n<\/div>\r\n<strong>Logically Equivalent Statements<\/strong>\r\n\r\nWhen the truth values for two statements are <strong>identical, <\/strong>the statements are said to be <strong>logically equivalent.\u00a0 \u00a0<\/strong>That is, both compositions of the same simple statements have the same meaning.\r\n<div class=\"textbox\">\r\n\r\n<strong>Common Equivalent Statements<\/strong>\r\n\r\n~(~p)\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0p\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<b><span style=\"color: c93344;\">the Double Negative Law<\/span><\/b>\r\n\r\np<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>q\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0q<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>p\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<b><span style=\"color: c93344;\">the Commutative Law for conjunction<\/span><\/b>\r\n\r\np<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>q\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0q<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>p\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<b><span style=\"color: c93344;\">the Commutative Law for disjunction<\/span><\/b>\r\n\r\n(p<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>q)<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>r\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0p<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>(q<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>r)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<b><span style=\"color: c93344;\">the Associative Law for conjunction<\/span><\/b>\r\n\r\n(p<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>q)<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>r\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0p<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>(q<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>r)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<b><span style=\"color: c93344;\">the Associative Law for disjunction<\/span><\/b>\r\n\r\n~(p<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>q)\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0(~p)<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>(~q)\u00a0 \u00a0 \u00a0 \u00a0<b><span style=\"color: c93344;\">DeMorgan's Laws\u00a0 (more on this one in the next section)<\/span><\/b>\r\n\r\n~(p<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>q)\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0(~p)<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>(~q)\r\n\r\np<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>(q<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>r)\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0(p<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>q)<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>(p<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>r) \u00a0 \u00a0<b><span style=\"color: c93344;\">the Distributive Laws<\/span><\/b>\r\n\r\np<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>(q<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>r)\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0(p<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>q)<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>(p<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>r)\r\n\r\np<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" \/>p\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0p\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<b><span style=\"color: c93344;\">Absorption Laws<\/span><\/b>\r\n\r\np<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" \/>p\u00a0<img src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" \/>\u00a0p\r\n\r\n<\/div>\r\nFor an example of logically equivalent, lets look at the forms of the conditional that we've learned.\u00a0 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>Conditional<\/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<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. Close this tab and proceed to the corresponding assignment.<\/span><\/strong><\/p>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>Interpreting Truth Tables<\/p>\n<ul>\n<li>Tautology<\/li>\n<li>Contradiction<\/li>\n<li>Contingency<\/li>\n<li>Logically Equivalent<\/li>\n<\/ul>\n<\/div>\n<p>A <strong>tautology<\/strong> is a proposition that is always true, regardless of the truth values of the propositional variables it contains. A proposition that is always false is called a <strong>contradiction<\/strong>. A proposition that is neither a tautology nor a contradiction is called a <strong>contingency<\/strong>.<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>An example of a <strong>tautology<\/strong> is:\u00a0 I am going to take Math for Liberal Arts this semester or I&#8217; m not going to take Math for Liberal Arts this semester.\u00a0 This statement is always true so it is a tautology.<\/p>\n<p>An example of a <strong>self-contradition<\/strong> is:\u00a0 I will get an A in this class and I will not get an A in this class.\u00a0 This statement is always false so it is a self-contradiction.<\/p>\n<p>&nbsp;<\/p>\n<p>Caution:\u00a0 Don&#8217;t make the mistake that every statement is either a tautology or a self-contradiction.\u00a0 We have seen many examples of truth tables which have a mixture of trues and falses in the final column.\u00a0 These statements are sometimes true and sometimes not true.\u00a0 These are called a <strong>contingency<\/strong>.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Complete a truth table to classify the statement as tautology, self-contradiction or contingency.<\/p>\n<p>a.\u00a0 p\u00a0<img decoding=\"async\" class=\"\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>\u00a0(~p)<\/p>\n<p>b.\u00a0 (p\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" width=\"8\" height=\"8\" alt=\"image\" \/>\u00a0q)\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" width=\"8\" height=\"8\" alt=\"image\" \/>\u00a0(~p\u00a0<img decoding=\"async\" class=\"\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>\u00a0~q)<\/p>\n<p>Solution:<\/p>\n<p>a.\u00a0 This is a tautology<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/03024926\/tautology.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6900\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/03024926\/tautology.png\" alt=\"\" width=\"164\" height=\"119\" \/><\/a><\/p>\n<p>b.\u00a0 This is a self-contradiction<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/03025325\/self-contradiction.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-6901\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/03025325\/self-contradiction-300x89.png\" alt=\"\" width=\"418\" height=\"124\" \/><\/a><\/p>\n<\/div>\n<p><strong>Logically Equivalent Statements<\/strong><\/p>\n<p>When the truth values for two statements are <strong>identical, <\/strong>the statements are said to be <strong>logically equivalent.\u00a0 \u00a0<\/strong>That is, both compositions of the same simple statements have the same meaning.<\/p>\n<div class=\"textbox\">\n<p><strong>Common Equivalent Statements<\/strong><\/p>\n<p>~(~p)\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0p\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<b><span style=\"color: c93344;\">the Double Negative Law<\/span><\/b><\/p>\n<p>p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>q\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0q<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>p\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<b><span style=\"color: c93344;\">the Commutative Law for conjunction<\/span><\/b><\/p>\n<p>p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>q\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0q<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>p\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<b><span style=\"color: c93344;\">the Commutative Law for disjunction<\/span><\/b><\/p>\n<p>(p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>q)<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>r\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>(q<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>r)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<b><span style=\"color: c93344;\">the Associative Law for conjunction<\/span><\/b><\/p>\n<p>(p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>q)<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>r\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>(q<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>r)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<b><span style=\"color: c93344;\">the Associative Law for disjunction<\/span><\/b><\/p>\n<p>~(p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>q)\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0(~p)<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>(~q)\u00a0 \u00a0 \u00a0 \u00a0<b><span style=\"color: c93344;\">DeMorgan&#8217;s Laws\u00a0 (more on this one in the next section)<\/span><\/b><\/p>\n<p>~(p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>q)\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0(~p)<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>(~q)<\/p>\n<p>p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>(q<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>r)\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0(p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>q)<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>(p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>r) \u00a0 \u00a0<b><span style=\"color: c93344;\">the Distributive Laws<\/span><\/b><\/p>\n<p>p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>(q<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>r)\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0(p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>q)<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>(p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>r)<\/p>\n<p>p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/smash.gif\" alt=\"image\" \/>p\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0p\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<b><span style=\"color: c93344;\">Absorption Laws<\/span><\/b><\/p>\n<p>p<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/logic\/wedge.gif\" alt=\"image\" \/>p\u00a0<img decoding=\"async\" src=\"https:\/\/www.zweigmedia.com\/RealWorld\/SYMB\/CN.GIF\" alt=\"image\" \/>\u00a0p<\/p>\n<\/div>\n<p>For an example of logically equivalent, lets look at the forms of the conditional that we&#8217;ve learned.\u00a0 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>Conditional<\/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<p style=\"text-align: center;\"><strong><span style=\"color: #ff0000;\">This is the end of the section. 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