What is the cardinal number of {1,2,4,8,16,\u2026,1024} and how do you find the number? Is there a formula for finding it?<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n The set {1,2,4,8,16,\u2026,1024}\\{1, 2, 4, 8, 16, \\dots, 1024\\} consists of powers of 2. The first element is 20=12^0 = 1, the second is 21=22^1 = 2, the third is 22=42^2 = 4, and so on. The last element in this sequence is 210=10242^{10} = 1024. This indicates that the set contains all the powers of 2 from 202^0 to 2102^{10}.<\/p>\n\n\n\n The cardinal number<\/strong> of a set is the number of elements it contains. To find the cardinal number of this set, we need to determine how many terms are included from 202^0 to 2102^{10}. These terms form an arithmetic progression where the first term is 202^0, and the common ratio is 2.<\/p>\n\n\n\n There are 11 terms in total: 20,21,22,\u2026,2102^0, 2^1, 2^2, \\dots, 2^{10}. Therefore, the cardinal number of the set is 11<\/strong>.<\/p>\n\n\n\n In general, for a sequence of powers of a number bb, from b0b^0 to bnb^n, the number of terms (cardinal number) can be found by the formula: Cardinality=n+1\\text{Cardinality} = n + 1<\/p>\n\n\n\n where nn is the exponent of the highest power in the sequence.<\/p>\n\n\n\n In our case, the sequence is 20,21,22,\u2026,2102^0, 2^1, 2^2, \\dots, 2^{10}, so the highest exponent is 10. Using the formula: Cardinality=10+1=11\\text{Cardinality} = 10 + 1 = 11<\/p>\n\n\n\n Thus, the cardinal number of the set {1,2,4,8,16,\u2026,1024}\\{1, 2, 4, 8, 16, \\dots, 1024\\} is 11.<\/p>\n\n\n\n This set consists of numbers that are successive powers of 2. The first element is 11, which is 202^0, and the last element is 10241024, which is 2102^{10}. The sequence includes all integers from 202^0 to 2102^{10}, and since these are powers of 2, they are distinct. Therefore, the total number of elements is equal to the number of terms in the sequence, which is 11.<\/p>\n","protected":false},"excerpt":{"rendered":" What is the cardinal number of {1,2,4,8,16,\u2026,1024} and how do you find the number? Is there a formula for finding it? The Correct Answer and Explanation is : The set {1,2,4,8,16,\u2026,1024}\\{1, 2, 4, 8, 16, \\dots, 1024\\} consists of powers of 2. The first element is 20=12^0 = 1, the second is 21=22^1 = 2, […]<\/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-186794","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186794","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=186794"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/186794\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=186794"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=186794"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=186794"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Cardinal Number:<\/h3>\n\n\n\n
Formula for Finding Cardinality:<\/h3>\n\n\n\n
Explanation:<\/h3>\n\n\n\n