{"id":184340,"date":"2025-01-20T19:09:00","date_gmt":"2025-01-20T19:09:00","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=184340"},"modified":"2025-01-20T19:09:03","modified_gmt":"2025-01-20T19:09:03","slug":"find-the-truth-set-of-each-predicate","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/20\/find-the-truth-set-of-each-predicate\/","title":{"rendered":"Find the truth set of each predicate"},"content":{"rendered":"\n<p>Find the truth set of each predicate. (If your answer is an interval, enter it using interval notation; otherwise enter it using set-roster notation.) 8 (a) Predicate: is an integer, domain: Z d {1,2,4,8) (b) Predicate: {1} 80 is an integer, domain: Z+ (c) Predicate: 1 = x\u00b2 = 4, domain: R [-2,-1] U <a href=\"d\">1,2<\/a> Predicate: 1 = x\u00b2 = 4, domain: Z {-1}<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>Here are the solutions and explanations for the truth sets of the given predicates:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(a) Predicate:<\/strong> x&nbsp;is&nbsp;an&nbsp;integerx \\text{ is an integer}, <strong>Domain:<\/strong> Z\u2216{1,2,4,8}\\mathbb{Z} \\setminus \\{1, 2, 4, 8\\}<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Solution:<\/strong><\/h4>\n\n\n\n<p>The predicate specifies that xx is an integer and must not belong to the set {1,2,4,8}\\{1, 2, 4, 8\\}. Since Z\\mathbb{Z} is the set of all integers and we exclude {1,2,4,8}\\{1, 2, 4, 8\\}, the truth set is: Z\u2216{1,2,4,8}={x\u2208Z\u2223x\u22601,x\u22602,x\u22604,x\u22608}.\\mathbb{Z} \\setminus \\{1, 2, 4, 8\\} = \\{x \\in \\mathbb{Z} \\mid x \\neq 1, x \\neq 2, x \\neq 4, x \\neq 8\\}.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Explanation:<\/strong><\/h4>\n\n\n\n<p>The domain excludes specific values {1,2,4,8}\\{1, 2, 4, 8\\}, while the predicate simply requires xx to be an integer. Therefore, the truth set includes all integers except those explicitly removed.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(b) Predicate:<\/strong> x=80&nbsp;and&nbsp;x&nbsp;is&nbsp;an&nbsp;integerx = 80 \\text{ and } x \\text{ is an integer}, <strong>Domain:<\/strong> Z+\\mathbb{Z}^+ (Positive integers)<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Solution:<\/strong><\/h4>\n\n\n\n<p>Since the predicate specifies x=80x = 80, the truth set depends on whether 8080 belongs to the domain Z+\\mathbb{Z}^+. Since Z+\\mathbb{Z}^+ represents all positive integers and 8080 is a positive integer, the truth set is: {80}.\\{80\\}.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Explanation:<\/strong><\/h4>\n\n\n\n<p>The predicate narrows the condition to a single value x=80x = 80, and 8080 satisfies both the conditions of being a positive integer and an integer.<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(c) Predicate:<\/strong> 1\u2264x2\u226441 \\leq x^2 \\leq 4, <strong>Domain:<\/strong> R\\mathbb{R}<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Solution:<\/strong><\/h4>\n\n\n\n<p>The inequality 1\u2264x2\u226441 \\leq x^2 \\leq 4 implies: x2\u2208[1,4].x^2 \\in [1, 4].<\/p>\n\n\n\n<p>Taking the square root, the solution for xx is: x\u2208[\u22122,\u22121]\u222a[1,2].x \\in [-2, -1] \\cup [1, 2].<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Explanation:<\/strong><\/h4>\n\n\n\n<p>For real numbers, x2\u22651x^2 \\geq 1 corresponds to values outside the interval (\u22121,1)(-1, 1), and x2\u22644x^2 \\leq 4 ensures the values remain within [\u22122,2][-2, 2]. Combining these gives x\u2208[\u22122,\u22121]\u222a[1,2]x \\in [-2, -1] \\cup [1, 2].<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>(d) Predicate:<\/strong> 1\u2264x2\u226441 \\leq x^2 \\leq 4, <strong>Domain:<\/strong> Z\u2216{\u22121}\\mathbb{Z} \\setminus \\{-1\\}<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Solution:<\/strong><\/h4>\n\n\n\n<p>For x2\u2208[1,4]x^2 \\in [1, 4] within the integers, xx must satisfy: x\u2208{\u22122,\u22121,1,2}.x \\in \\{-2, -1, 1, 2\\}.<\/p>\n\n\n\n<p>Excluding \u22121-1 from the domain, the truth set becomes: {\u22122,1,2}.\\{-2, 1, 2\\}.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>Explanation:<\/strong><\/h4>\n\n\n\n<p>The condition x2\u2208[1,4]x^2 \\in [1, 4] provides the integer solutions {\u22122,\u22121,1,2}\\{-2, -1, 1, 2\\}. However, \u22121-1 is excluded, leaving {\u22122,1,2}\\{-2, 1, 2\\} as the truth set.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find the truth set of each predicate. (If your answer is an interval, enter it using interval notation; otherwise enter it using set-roster notation.) 8 (a) Predicate: is an integer, domain: Z d {1,2,4,8) (b) Predicate: {1} 80 is an integer, domain: Z+ (c) Predicate: 1 = x\u00b2 = 4, domain: R [-2,-1] U 1,2 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-184340","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184340","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=184340"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184340\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=184340"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=184340"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=184340"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}