{"id":423,"date":"2022-04-13T10:57:20","date_gmt":"2022-04-13T10:57:20","guid":{"rendered":"http:\/\/10thclass.deltapublications.in\/?page_id=423"},"modified":"2025-04-05T09:12:12","modified_gmt":"2025-04-05T09:12:12","slug":"w-2-counterexamples","status":"publish","type":"page","link":"https:\/\/10thclass.deltapublications.in\/index.php\/w-2-counterexamples\/","title":{"rendered":"W.2 Counterexamples"},"content":{"rendered":"\n<h2 class=\"wp-block-heading has-text-align-center has-text-color\" style=\"color:#00056d;text-transform:uppercase\"><strong> Counterexamples<\/strong><\/h2>\n\n\n\n<p class=\"has-text-color has-link-color has-huge-font-size wp-elements-f79afd6ea8b322268468366b8e5f7cc7\" style=\"color:#74008b\"><strong>Key Notes:<\/strong><\/p>\n\n\n\n<p class=\"has-large-font-size\"> \ud83c\udf88 A conditional statement can be expressed as&nbsp;<strong>If&nbsp;<em>A<\/em>, then&nbsp;<em>B<\/em><\/strong>.<\/p>\n\n\n\n<p class=\"has-large-font-size\">\ud83c\udf88 <em>A<\/em>&nbsp;is the&nbsp;<strong>hypothesis<\/strong>&nbsp;and&nbsp;<em>B<\/em>&nbsp;is the&nbsp;<strong>conclusion<\/strong>.<\/p>\n\n\n\n<p class=\"has-large-font-size\">\ud83c\udf88 A&nbsp;<strong>counterexample<\/strong>&nbsp;is an example in which the hypothesis is true, but the conclusion is false. If you can find a counterexample to a conditional statement, then that conditional statement is false.<\/p>\n\n\n\n<p class=\"has-text-align-center has-text-color has-large-font-size\" style=\"color:#105000\"><strong>Learn with an example<\/strong><\/p>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-large-font-size\"><div class=\"wp-block-group__inner-container is-layout-constrained wp-block-group-is-layout-constrained\">\n<div class=\"wp-block-group has-background\" style=\"background-color:#e1efc0\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color has-link-color wp-elements-1d9e8aad215a9b9307c16aa7bed0fbbf\" style=\"color:#b00012\"> \ud83d\udeec <strong>Which of the following is a counter<\/strong><\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-4009cdabcca607e1e09248953ed31a01\" style=\"color:#b00012\"><strong>example for this conditional statement?<\/strong><\/p>\n\n\n\n<p>\ud83d\udeec If&nbsp;a child is a girl,&nbsp;then&nbsp;the child&#8217;s favourite colour is pink.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Ben, a boy whose favourite colour is pink<\/li>\n\n\n\n<li>Kevin, a boy whose favourite colour is blue<\/li>\n\n\n\n<li>Maura, a girl whose favourite colour is pink<\/li>\n\n\n\n<li>Audrey, a girl whose favourite colour is green<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First notice that the given statement is in&nbsp;if\/then&nbsp;form:<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>If&nbsp;<\/td><td><strong>a child is a girl<\/strong>,&nbsp;<\/td><td>then&nbsp;<\/td><td><strong>the child&#8217;s favourite colour is pink<\/strong>.&nbsp;<\/td><\/tr><tr><td>If&nbsp;<\/td><td><strong><em>A<\/em><\/strong>,&nbsp;<\/td><td>then&nbsp;<\/td><td><strong><em>B<\/em><\/strong>.&nbsp;<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Now go through the answer choices one by one, looking for the counterexample that makes the&nbsp;<strong>hypothesis<\/strong>&nbsp;true and the&nbsp;<strong>conclusion<\/strong>&nbsp;false.<\/li>\n\n\n\n<li>First try&nbsp;<strong>Ben, a boy whose favourite colour is pink<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is false, because this child is not a girl. A counterexample must make the hypothesis true, so this is not a counterexample.<\/li>\n\n\n\n<li>Next try&nbsp;<strong>Kevin, a boy whose favourite colour is blue<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is false, because this child is not a girl. A counterexample must make the hypothesis true, so this is not a counterexample.<\/li>\n\n\n\n<li>Next try&nbsp;<strong>Maura, a girl whose favourite colour is pink<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is true, because this child is a girl. The&nbsp;<strong>conclusion<\/strong>&nbsp;is also true, because this child&#8217;s favourite colour is pink. A counterexample must make the conclusion false, so this is not a counterexample.<\/li>\n\n\n\n<li>Finally try&nbsp;<strong>Audrey, a girl whose favourite colour is green<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is true, because this child is a girl. The&nbsp;<strong>conclusion<\/strong>&nbsp;is false, because this child&#8217;s favourite colour is not pink. A counterexample makes the hypothesis true and the conclusion false. So, this is a counterexample.<\/li>\n\n\n\n<li>In summary, the counterexample is&nbsp;<strong>Audrey, a girl whose favourite colour is green<\/strong>.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#e6f8ff\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p> \ud83d\udeec Which of the following is a counterexample for this conditional statement?<\/p>\n\n\n\n<div class=\"wp-block-group\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-text-color has-link-color wp-elements-1f069d23dafef4c22eb17240773a27c1\" style=\"color:#b00012\"><strong>\ud83d\udeec If&nbsp;the sum of two integers is even,&nbsp;then&nbsp;those integers are both even.&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>8 and 1<\/li>\n\n\n\n<li>2 and 4<\/li>\n\n\n\n<li>6 and 4<\/li>\n\n\n\n<li>1 and 3<\/li>\n<\/ul>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First notice that the given statement is in&nbsp;if\/then&nbsp;form:<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>If&nbsp;<\/td><td><strong>the sum of two integers is even<\/strong>,&nbsp;<\/td><td>then&nbsp;<\/td><td><strong>those integers are both even<\/strong>.&nbsp;<\/td><\/tr><tr><td>If&nbsp;<\/td><td><strong><em>A<\/em><\/strong>,&nbsp;<\/td><td>then&nbsp;<\/td><td><strong><em>B<\/em><\/strong>.&nbsp;<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Now go through the answer choices one by one, looking for the counterexample that makes the&nbsp;<strong>hypothesis<\/strong>&nbsp;true and the&nbsp;<strong>conclusion<\/strong>&nbsp;false.<\/li>\n\n\n\n<li>First try&nbsp;<strong>8 and 1<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is false, because the sum of 8 and 1 is not even. A counterexample must make the hypothesis true, so this is not a counterexample.<\/li>\n\n\n\n<li>Next try&nbsp;<strong>2 and 4<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is true, because the sum of 2 and 4 is even. The&nbsp;<strong>conclusion<\/strong>&nbsp;is also true, because 2 and 4 are both even. A counterexample must make the conclusion false, so this is not a counterexample.<\/li>\n\n\n\n<li>Next try&nbsp;<strong>6 and 4<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is true, because the sum of 6 and 4 is even. The&nbsp;<strong>conclusion<\/strong>&nbsp;is also true, because 6 and 4 are both even. A counterexample must make the conclusion false, so this is not a counterexample.<\/li>\n\n\n\n<li>Finally try&nbsp;<strong>1 and 3<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is true, because the sum of 1 and 3 is even. The&nbsp;<strong>conclusion<\/strong>&nbsp;is false, because 1 and 3 are not both even. A counterexample makes the hypothesis true and the conclusion false. So, this is a counterexample.<\/li>\n\n\n\n<li>In summary, the counterexample is&nbsp;<strong>1 and 3<\/strong>.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-group has-background\" style=\"background-color:#fbddf2\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<div class=\"wp-block-group has-background-background-color has-background\"><div class=\"wp-block-group__inner-container is-layout-flow wp-block-group-is-layout-flow\">\n<p class=\"has-primary-color has-text-color has-link-color wp-elements-ec38d595a90a0b6d0caed28c084cee4d\"> \ud83d\udeecWhich of the following is a counterexample for this conditional statement?<\/p>\n\n\n\n<p class=\"has-text-color has-link-color wp-elements-c682bbf9456824d54ed224fc9ad8cdd9\" style=\"color:#b00012\"><strong> \ud83d\udeec If&nbsp;an animal walks on two legs,&nbsp;then&nbsp;it is a human.&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>an elephant<\/li>\n\n\n\n<li>a giraffe<\/li>\n\n\n\n<li>a kangaroo<\/li>\n\n\n\n<li>a dolphin<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>First notice that the given statement is in&nbsp;if\/then&nbsp;form:<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-table\"><table><tbody><tr><td>If&nbsp;<\/td><td><strong>an animal walks on two legs<\/strong>,&nbsp;<\/td><td>then&nbsp;<\/td><td><strong>it is a human<\/strong>.&nbsp;<\/td><\/tr><tr><td>If&nbsp;<\/td><td><strong><em>A<\/em><\/strong>,&nbsp;<\/td><td>then&nbsp;<\/td><td><strong><em>B<\/em><\/strong>.&nbsp;<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<ul class=\"wp-block-list has-large-font-size\">\n<li>Now go through the answer choices one by one, looking for the counterexample that makes the&nbsp;<strong>hypothesis<\/strong>&nbsp;true and the&nbsp;<strong>conclusion<\/strong>&nbsp;false.<\/li>\n\n\n\n<li>First try&nbsp;<strong>an elephant<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is false, because an elephant does not walk on two legs. A counterexample must make the hypothesis true, so this is not a counterexample.<\/li>\n\n\n\n<li>Next try&nbsp;<strong>a giraffe<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is false, because a giraffe does not walk on two legs. A counterexample must make the hypothesis true, so this is not a counterexample.<\/li>\n\n\n\n<li>Next try&nbsp;<strong>a kangaroo<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is true, because a kangaroo walks on two legs. The&nbsp;<strong>conclusion<\/strong>&nbsp;is false, because a kangaroo is not a human. A counterexample makes the hypothesis true and the conclusion false. So, this is a counterexample.<\/li>\n\n\n\n<li>Finally try&nbsp;<strong>a dolphin<\/strong>.&nbsp;The&nbsp;<strong>hypothesis<\/strong>&nbsp;is false, because a dolphin does not walk on two legs. A counterexample must make the hypothesis true, so this is not a counterexample.<\/li>\n\n\n\n<li>In summary, the counterexample is&nbsp;<strong>a kangaroo<\/strong>.<\/li>\n<\/ul>\n<\/div><\/div>\n\n\n\n<p class=\"has-text-color has-large-font-size\" style=\"color:#8b0b0b\">Let&#8217;s practice!\ud83d\udd8a\ufe0f <\/p>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-columns is-layout-flex wp-container-core-columns-is-layout-9d6595d7 wp-block-columns-is-layout-flex\">\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/89891\/892\/389\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-152.png\" alt=\"\" class=\"wp-image-7456\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-152.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-152-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-3-152-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n\n\n\n<div class=\"wp-block-column is-layout-flow wp-block-column-is-layout-flow\">\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/wordwall.net\/play\/73565\/257\/588\"><img loading=\"lazy\" decoding=\"async\" width=\"500\" height=\"500\" src=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-171.png\" alt=\"\" class=\"wp-image-7457\" srcset=\"https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-171.png 500w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-171-300x300.png 300w, https:\/\/10thclass.deltapublications.in\/wp-content\/uploads\/2023\/05\/Worksheet-1-1-2-171-150x150.png 150w\" sizes=\"auto, (max-width: 500px) 100vw, 500px\" \/><\/a><\/figure>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Counterexamples Key Notes: \ud83c\udf88 A conditional statement can be expressed as&nbsp;If&nbsp;A, then&nbsp;B. \ud83c\udf88 A&nbsp;is the&nbsp;hypothesis&nbsp;and&nbsp;B&nbsp;is the&nbsp;conclusion. \ud83c\udf88 A&nbsp;counterexample&nbsp;is an example in which the hypothesis is true, but the conclusion is false. If you can find a counterexample to a conditional statement, then that conditional statement is false. Learn with an example \ud83d\udeec Which of the<a class=\"more-link\" href=\"https:\/\/10thclass.deltapublications.in\/index.php\/w-2-counterexamples\/\">Continue reading <span class=\"screen-reader-text\">&#8220;W.2 Counterexamples&#8221;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"class_list":["post-423","page","type-page","status-publish","hentry","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/423","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/comments?post=423"}],"version-history":[{"count":12,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/423\/revisions"}],"predecessor-version":[{"id":17665,"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/pages\/423\/revisions\/17665"}],"wp:attachment":[{"href":"https:\/\/10thclass.deltapublications.in\/index.php\/wp-json\/wp\/v2\/media?parent=423"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}