{"id":6791,"date":"2021-12-30T21:42:16","date_gmt":"2021-12-30T21:42:16","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/?post_type=chapter&#038;p=6791"},"modified":"2022-12-14T18:01:12","modified_gmt":"2022-12-14T18:01:12","slug":"1-8-truth-tables-conditionals-and-biconditionals","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-mathforliberalartscorequisite1\/chapter\/1-8-truth-tables-conditionals-and-biconditionals\/","title":{"raw":"1.8 Truth Tables: Conditionals and Biconditionals","rendered":"1.8 Truth Tables: Conditionals and Biconditionals"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3 style=\"padding-left: 30px;\">Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Basic Truth Tables for\r\n<ul>\r\n \t<li>Conditional<\/li>\r\n \t<li>Biconditional<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Working with the Conditional Statement\r\n<ul>\r\n \t<li>Converse<\/li>\r\n \t<li>Inverse<\/li>\r\n \t<li>Contrapositive<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox\"><strong>Conditional<\/strong>\r\nThis is sometimes called an implication. A conditional is a logical compound statement in which a statement p, called the antecedent, implies a statement q, called the consequent.\r\nA conditional is written as p \u2192 q and is translated as \u201cif p, then q\u201d.<\/div>\r\nThe English statement \u201cIf it is raining, then there are clouds is the sky\u201d is a conditional statement. It makes sense because if the antecedent \u201cit is raining\u201d is true, then the consequent \u201cthere are clouds in the sky\u201d must also be true.\r\n\r\nNotice that the statement tells us nothing of what to expect if it is not raining; there might be\r\nclouds in the sky, or there might not. If the antecedent is false, then the consequent becomes\r\nirrelevant.\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nSuppose you order a team jersey online on Tuesday and want to receive it by Friday so you can\r\nwear it to Saturday\u2019s game. The website says that if you pay for expedited shipping, you will\r\nreceive the jersey by Friday. In what situation is the website telling a lie?\r\nThere are four possible outcomes:\r\n1) You pay for expedited shipping and receive the jersey by Friday\r\n2) You pay for expedited shipping and don\u2019t receive the jersey by Friday\r\n3) You don\u2019t pay for expedited shipping and receive the jersey by Friday\r\n4) You don\u2019t pay for expedited shipping and don\u2019t receive the jersey by Friday\r\n\r\nOnly one of these outcomes proves that the website was lying: the second outcome in which you pay for expedited shipping but don\u2019t receive the jersey by Friday. The first outcome is exactly what was promised, so there\u2019s no problem with that. The third outcome is not a lie because the website never said what would happen if you didn\u2019t pay for expedited shipping;\r\nmaybe the jersey would arrive by Friday whether you paid for expedited shipping or not. The fourth outcome is not a lie because, again, the website didn\u2019t make any promises about when the jersey would arrive if you didn\u2019t pay for expedited shipping.\r\n\r\nIt may seem strange that the third outcome in the previous example, in which the first part is false but the second part is true, is not a lie. Remember, though, that if the antecedent is false, we cannot make any judgment about the consequent. The website never said that paying for expedited shipping was the only way to receive the jersey by Friday.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Example<\/h3>\r\nA friend tells you \u201cIf you upload that picture to Facebook, you\u2019ll lose your job.\u201d Under what conditions can you say that your friend was wrong?\r\nThere are four possible outcomes:\r\n1) You upload the picture and lose your job\r\n2) You upload the picture and don\u2019t lose your job\r\n3) You don\u2019t upload the picture and lose your job\r\n4) You don\u2019t upload the picture and don\u2019t lose your job\r\nThere is only one possible case in which you can say your friend was wrong: the second outcome in which you upload the picture but still 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 say that their statement was wrong. Even if you didn\u2019t upload the picture and lost your job anyway, your\r\nfriend never said that you were guaranteed to keep your job if you didn\u2019t upload the picture; you might lose your job for missing a shift or punching your boss instead.\r\n\r\n<\/div>\r\nA\u00a0<b>conditional statement<\/b>\u00a0tells us that if the antecedent is true, the consequent cannot be false. Thus, a conditional statement is only false when a true antecedent implies a false consequent.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/02170729\/Conditional.png\"><img class=\"size-medium wp-image-6889 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/02170729\/Conditional-300x216.png\" alt=\"\" width=\"300\" height=\"216\" \/><\/a>\r\n\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nAnother example is living in an apartment and paying rent.\u00a0p \u2192 q where p is I live in an apartment and q is then I pay rent.\u00a0 What are the outcomes?\r\n<ol>\r\n \t<li>I do live in an apartment and I pay rent, then the situation is true (no eviction!)<\/li>\r\n \t<li>I live in an apartment and I don't pay rent, then the situation is false (eviction, broken promise)<\/li>\r\n \t<li>I don't live in an apartment but I do pay rent, then the situation is true (though why would you do it?)<\/li>\r\n \t<li>I don't live in an apartment and I don't pay rent, then the situation is true (no promise broken)<\/li>\r\n<\/ol>\r\n<\/div>\r\nWith conditional situations, we also have the following:\r\n<div class=\"textbox\"><strong>Related Statements<\/strong>\r\nThe original conditional is \u201cif p, then q\u201d p \u2192 q\r\nThe<strong> converse<\/strong> is \u201cif q, then p\u201d q \u2192 p\r\nThe <strong>inverse<\/strong> is \u201cif not p, then not q\u201d ~p \u2192~ q\r\nThe <strong>contrapositive<\/strong> is \u201cif not q, then not p\u201d ~q \u2192 ~p<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nConsider the conditional \u201cIf it is raining, then there are clouds in the sky.\u201d It seems reasonable to assume that this is true.\r\nThe <strong>converse<\/strong> would be \u201cIf there are clouds in the sky, then it is raining.\u201d This is not always true.\r\nThe <strong>inverse<\/strong> would be \u201cIf it is not raining, then there are not clouds in the sky.\u201d Likewise, this is not always true.\r\nThe <strong>contrapositive<\/strong> would be \u201cIf there are not clouds in the sky, then it is not raining.\u201d This\r\nstatement is true, and is equivalent to the original conditional.\r\n\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nSuppose this statement is true: \u201cIf I eat this giant cookie, then I will feel sick.\u201d Which of the following statements must also be true?\r\na. If I feel sick, then I ate that giant cookie.\r\nb. If I don\u2019t eat this giant cookie, then I won\u2019t feel sick.\r\nc. If I don\u2019t feel sick, then I didn\u2019t eat that giant cookie.\r\n\r\na. This is the converse, which is not necessarily true. I could feel sick for some other reason, such as drinking sour milk.\r\n\r\nb. This is the inverse, which is not necessarily true. Again, I could feel sick for some other reason; avoiding the cookie doesn\u2019t guarantee that I won\u2019t feel sick.\r\n\r\nc. This is the contrapositive, which is true, but we have to think somewhat backwards to explain it. If I ate the cookie, I would feel sick, but since I don\u2019t feel sick, I must not have eaten the cookie.\r\nNotice again that the original statement and the contrapositive have the same truth value (both\r\nare true), and the converse and the inverse have the same truth value (both are false).\r\n\r\n<\/div>\r\n<div class=\"textbox\"><strong>Biconditional<\/strong>\r\nA biconditional is a logical conditional statement in which the antecedent and consequent are interchangeable.\r\nA biconditional is written as p \u2194 q and is translated as \u201cp if and only if q\u201d.<\/div>\r\nBecause a biconditional statement p \u2194 q is equivalent to (p \u2192 q) \u22c0 (q \u2192 p), we may think of it as a conditional statement combined with its converse: if p, then q and if q, then p. The double-headed arrow shows that the conditional statement goes from left to right and from right to left. A biconditional is considered true as long as the antecedent and the consequent have the same truth value; that is, they are either both true or both false.\r\n\r\nThe\u00a0<b>biconditional<\/b>\u00a0tells us that, \u201cEither both are the case, or neither is\u2026 \u201d Thus, a biconditional statement is true when both statements are true, or both are false.\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/02171232\/Biconditional.png\"><img class=\"size-medium wp-image-6891 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/02171232\/Biconditional-300x215.png\" alt=\"\" width=\"300\" height=\"215\" \/><\/a>\r\n\r\n&nbsp;\r\n<div class=\"textbox examples\">\r\n<h3>Examples<\/h3>\r\nSuppose this statement is true: \u201cThe garbage truck comes down my street if and only if it is Thursday morning.\u201d Which of the following statements could be true?\r\na. It is noon on Thursday and the garbage truck did not come down my street this morning.\r\nb. It is Monday and the garbage truck is coming down my street.\r\nc. It is Wednesday at 11:59PM and the garbage truck did not come down my street today.\r\n\r\n&nbsp;\r\n\r\na. This cannot be true. This is like the second row of the truth table; it is true that I just experienced Thursday morning, but it is false that the garbage truck came.\r\nb. This cannot be true. This is like the third row of the truth table; it is false that it is Thursday, but it is true that the garbage truck came.\r\nc. This could be true. This is like the fourth row of the truth table; it is false that it is Thursday, but it is also false that the garbage truck came, so everything worked out like it should.\r\n\r\n<\/div>\r\n<strong>Working with the Conditional Statement<\/strong>\r\n<p style=\"padding-left: 30px;\">Conditional statements play a very big role in logic and one of the ways we can learn more about them is to study the three related statements.<\/p>\r\n\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\n&nbsp;\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;\">\u00a0This 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 style=\"padding-left: 30px;\">Learning Objectives<\/h3>\n<ul>\n<li>Basic Truth Tables for\n<ul>\n<li>Conditional<\/li>\n<li>Biconditional<\/li>\n<\/ul>\n<\/li>\n<li>Working with the Conditional Statement\n<ul>\n<li>Converse<\/li>\n<li>Inverse<\/li>\n<li>Contrapositive<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\"><strong>Conditional<\/strong><br \/>\nThis is sometimes called an implication. A conditional is a logical compound statement in which a statement p, called the antecedent, implies a statement q, called the consequent.<br \/>\nA conditional is written as p \u2192 q and is translated as \u201cif p, then q\u201d.<\/div>\n<p>The English statement \u201cIf it is raining, then there are clouds is the sky\u201d is a conditional statement. It makes sense because if the antecedent \u201cit is raining\u201d is true, then the consequent \u201cthere are clouds in the sky\u201d must also be true.<\/p>\n<p>Notice that the statement tells us nothing of what to expect if it is not raining; there might be<br \/>\nclouds in the sky, or there might not. If the antecedent is false, then the consequent becomes<br \/>\nirrelevant.<\/p>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>Suppose you order a team jersey online on Tuesday and want to receive it by Friday so you can<br \/>\nwear it to Saturday\u2019s game. The website says that if you pay for expedited shipping, you will<br \/>\nreceive the jersey by Friday. In what situation is the website telling a lie?<br \/>\nThere are four possible outcomes:<br \/>\n1) You pay for expedited shipping and receive the jersey by Friday<br \/>\n2) You pay for expedited shipping and don\u2019t receive the jersey by Friday<br \/>\n3) You don\u2019t pay for expedited shipping and receive the jersey by Friday<br \/>\n4) You don\u2019t pay for expedited shipping and don\u2019t receive the jersey by Friday<\/p>\n<p>Only one of these outcomes proves that the website was lying: the second outcome in which you pay for expedited shipping but don\u2019t receive the jersey by Friday. The first outcome is exactly what was promised, so there\u2019s no problem with that. The third outcome is not a lie because the website never said what would happen if you didn\u2019t pay for expedited shipping;<br \/>\nmaybe the jersey would arrive by Friday whether you paid for expedited shipping or not. The fourth outcome is not a lie because, again, the website didn\u2019t make any promises about when the jersey would arrive if you didn\u2019t pay for expedited shipping.<\/p>\n<p>It may seem strange that the third outcome in the previous example, in which the first part is false but the second part is true, is not a lie. Remember, though, that if the antecedent is false, we cannot make any judgment about the consequent. The website never said that paying for expedited shipping was the only way to receive the jersey by Friday.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Example<\/h3>\n<p>A friend tells you \u201cIf you upload that picture to Facebook, you\u2019ll lose your job.\u201d Under what conditions can you say that your friend was wrong?<br \/>\nThere are four possible outcomes:<br \/>\n1) You upload the picture and lose your job<br \/>\n2) You upload the picture and don\u2019t lose your job<br \/>\n3) You don\u2019t upload the picture and lose your job<br \/>\n4) You don\u2019t upload the picture and don\u2019t lose your job<br \/>\nThere is only one possible case in which you can say your friend was wrong: the second outcome in which you upload the picture but still 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 say that their statement was wrong. Even if you didn\u2019t upload the picture and lost your job anyway, your<br \/>\nfriend never said that you were guaranteed to keep your job if you didn\u2019t upload the picture; you might lose your job for missing a shift or punching your boss instead.<\/p>\n<\/div>\n<p>A\u00a0<b>conditional statement<\/b>\u00a0tells us that if the antecedent is true, the consequent cannot be false. Thus, a conditional statement is only false when a true antecedent implies a false consequent.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/02170729\/Conditional.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-6889 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/02170729\/Conditional-300x216.png\" alt=\"\" width=\"300\" height=\"216\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Another example is living in an apartment and paying rent.\u00a0p \u2192 q where p is I live in an apartment and q is then I pay rent.\u00a0 What are the outcomes?<\/p>\n<ol>\n<li>I do live in an apartment and I pay rent, then the situation is true (no eviction!)<\/li>\n<li>I live in an apartment and I don&#8217;t pay rent, then the situation is false (eviction, broken promise)<\/li>\n<li>I don&#8217;t live in an apartment but I do pay rent, then the situation is true (though why would you do it?)<\/li>\n<li>I don&#8217;t live in an apartment and I don&#8217;t pay rent, then the situation is true (no promise broken)<\/li>\n<\/ol>\n<\/div>\n<p>With conditional situations, we also have the following:<\/p>\n<div class=\"textbox\"><strong>Related Statements<\/strong><br \/>\nThe original conditional is \u201cif p, then q\u201d p \u2192 q<br \/>\nThe<strong> converse<\/strong> is \u201cif q, then p\u201d q \u2192 p<br \/>\nThe <strong>inverse<\/strong> is \u201cif not p, then not q\u201d ~p \u2192~ q<br \/>\nThe <strong>contrapositive<\/strong> is \u201cif not q, then not p\u201d ~q \u2192 ~p<\/div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Consider the conditional \u201cIf it is raining, then there are clouds in the sky.\u201d It seems reasonable to assume that this is true.<br \/>\nThe <strong>converse<\/strong> would be \u201cIf there are clouds in the sky, then it is raining.\u201d This is not always true.<br \/>\nThe <strong>inverse<\/strong> would be \u201cIf it is not raining, then there are not clouds in the sky.\u201d Likewise, this is not always true.<br \/>\nThe <strong>contrapositive<\/strong> would be \u201cIf there are not clouds in the sky, then it is not raining.\u201d This<br \/>\nstatement is true, and is equivalent to the original conditional.<\/p>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Suppose this statement is true: \u201cIf I eat this giant cookie, then I will feel sick.\u201d Which of the following statements must also be true?<br \/>\na. If I feel sick, then I ate that giant cookie.<br \/>\nb. If I don\u2019t eat this giant cookie, then I won\u2019t feel sick.<br \/>\nc. If I don\u2019t feel sick, then I didn\u2019t eat that giant cookie.<\/p>\n<p>a. This is the converse, which is not necessarily true. I could feel sick for some other reason, such as drinking sour milk.<\/p>\n<p>b. This is the inverse, which is not necessarily true. Again, I could feel sick for some other reason; avoiding the cookie doesn\u2019t guarantee that I won\u2019t feel sick.<\/p>\n<p>c. This is the contrapositive, which is true, but we have to think somewhat backwards to explain it. If I ate the cookie, I would feel sick, but since I don\u2019t feel sick, I must not have eaten the cookie.<br \/>\nNotice again that the original statement and the contrapositive have the same truth value (both<br \/>\nare true), and the converse and the inverse have the same truth value (both are false).<\/p>\n<\/div>\n<div class=\"textbox\"><strong>Biconditional<\/strong><br \/>\nA biconditional is a logical conditional statement in which the antecedent and consequent are interchangeable.<br \/>\nA biconditional is written as p \u2194 q and is translated as \u201cp if and only if q\u201d.<\/div>\n<p>Because a biconditional statement p \u2194 q is equivalent to (p \u2192 q) \u22c0 (q \u2192 p), we may think of it as a conditional statement combined with its converse: if p, then q and if q, then p. The double-headed arrow shows that the conditional statement goes from left to right and from right to left. A biconditional is considered true as long as the antecedent and the consequent have the same truth value; that is, they are either both true or both false.<\/p>\n<p>The\u00a0<b>biconditional<\/b>\u00a0tells us that, \u201cEither both are the case, or neither is\u2026 \u201d Thus, a biconditional statement is true when both statements are true, or both are false.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/02171232\/Biconditional.png\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-6891 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5548\/2021\/12\/02171232\/Biconditional-300x215.png\" alt=\"\" width=\"300\" height=\"215\" \/><\/a><\/p>\n<p>&nbsp;<\/p>\n<div class=\"textbox examples\">\n<h3>Examples<\/h3>\n<p>Suppose this statement is true: \u201cThe garbage truck comes down my street if and only if it is Thursday morning.\u201d Which of the following statements could be true?<br \/>\na. It is noon on Thursday and the garbage truck did not come down my street this morning.<br \/>\nb. It is Monday and the garbage truck is coming down my street.<br \/>\nc. It is Wednesday at 11:59PM and the garbage truck did not come down my street today.<\/p>\n<p>&nbsp;<\/p>\n<p>a. This cannot be true. This is like the second row of the truth table; it is true that I just experienced Thursday morning, but it is false that the garbage truck came.<br \/>\nb. This cannot be true. This is like the third row of the truth table; it is false that it is Thursday, but it is true that the garbage truck came.<br \/>\nc. This could be true. This is like the fourth row of the truth table; it is false that it is Thursday, but it is also false that the garbage truck came, so everything worked out like it should.<\/p>\n<\/div>\n<p><strong>Working with the Conditional Statement<\/strong><\/p>\n<p style=\"padding-left: 30px;\">Conditional statements play a very big role in logic and one of the ways we can learn more about them is to study the three related statements.<\/p>\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>&nbsp;<\/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;\">\u00a0This is the end of the section. 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