উপসেট (Subset) (৭.৫)

অষ্টম শ্রেণি (মাধ্যমিক ২০২৫) - গণিত সেট | - | NCTB BOOK
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মনে করি, A = {a, b} একটি সেট। A সেটের উপাদান নিয়ে আমরা {a, b}, {a}, {b} সেটগুলো গঠন করতে পারি। গঠিত {a, b}, {a}, {b} সেটগুলো A সেটের উপসেট। কোনো সেটের উপাদান থেকে যতগুলো সেট গঠন করা যায় এদের প্রত্যেকটি প্রদত্ত সেটের উপসেট।

যেমন : P = {2, 3, 4, 5} এবং Q = {3,5} হলে, Q সেটটি P সেটের উপসেট। অর্থাৎ Q ⊆ P. কারণ Q সেটের 3 এবং 5 উপাদানসমূহ P সেটে বিদ্যমান। '2' প্রতীক দ্বারা উপসেটকে সূচিত করা হয়।

উদাহরণ ৪। A = {1, 2, 3} এর উপসেটসমূহ লেখ।

সমাধান : A সেটের উপসেটসমূহ নিম্নরূপ :

{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, Ø

 

সার্বিক সেট (Universal Set)

আলোচনায় সংশ্লিষ্ট সকল সেট যদি একটি নির্দিষ্ট সেটের উপসেট হয় তবে ঐ নির্দিষ্ট সেটকে এর উপসেটগুলোর সাপেক্ষে সার্বিক সেট বলে। সার্বিক সেটকে U প্রতীক দ্বারা সূচিত করা হয়। যেমন: কোনো বিদ্যালয়ের সকল শিক্ষার্থীর সেট হলো সার্বিক সেট এবং অষ্টম শ্রেণির শিক্ষার্থীদের সেট উক্ত সার্বিক সেটের উপসেট।

সকল সেট সার্বিক সেটের উপসেট।

উদাহরণ ৫। A = {1, 2, 3, 4, 5, 6}, B = {1, 3, 5,} C = { 3, 4, 5, 6} হলে, সার্বিক সেট নির্ণয় কর।

সমাধান : দেওয়া আছে, A = {1, 2, 3, 4, 5, 6}, B = {1, 3, 5, C = { 3, 4, 5, 6, }

এখানে, B সেটের উপাদান 1, 3, 5 এবং C সেটের উপাদান 3, 4, 5, 6 যা A সেটে বিদ্যমান। 

∴ B এবং C সেটের সাপেক্ষে সার্বিক সেট A

 

পূরক সেট (Complement of a set)

যদি U সার্বিক সেট এবং A সেটটি U এর উপসেট হয় তবে, A সেটের বহির্ভূত সকল উপাদান নিয়ে যে সেট গঠন করা হয়, একে A সেটের পূরক সেট বলে। A এর পূরক সেটকে A বা A' দ্বারা প্রকাশ করা হয়।

মনে করি, অষ্টম শ্রেণির 60 জন শিক্ষার্থীর মধ্যে 9 জন অনুপস্থিত। অষ্টম শ্রেণির সকল শিক্ষার্থীদের সেট সার্বিক সেট বিবেচনা করলে উপস্থিত (60–9) বা 51 জনের সেটের পূরক সেট হবে অনুপস্থিত 9 জনের সেট।

 

উদাহরণ ৬। U = {1, 2, 3, 4, 5, 6} এবং A = {2, 4, 6} হলে A নির্ণয় কর।

সমাধান : দেওয়া আছে, U = {1, 2, 3, 4, 5, 6} এবং A = {2, 4, 6,}

∴ Ac = A এর পূরক সেট

        = A এর বহির্ভূত উপাদানসমূহের সেট

        = {1, 3, 5}

কাজ : A = {a, b, c} হলে, 4 এর উপসেটসমূহ নির্ণয় কর এবং যেকোনো তিনটি উপসেট লিখে এদের পূরক সেট নির্ণয় কর
common.content_added_and_updated_by
টপ রেটেড অ্যাপ

স্যাট অ্যাকাডেমী অ্যাপ

আমাদের অল-ইন-ওয়ান মোবাইল অ্যাপের মাধ্যমে সীমাহীন শেখার সুযোগ উপভোগ করুন।

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Request history8.2.28PHP Version93.06msRequest Duration9MBMemory UsageGET academy/{slug}Route
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      uri
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        • app/Services/SubjectService.php:32
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        • app/Services/SubjectService.php:43
        • app/Http/Controllers/Academy/SubjectController.php:105
        • vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
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        • vendor/kalnoy/nestedset/src/QueryBuilder.php:38
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        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
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        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
      • QueryBuilder.php#305satt_satt_mobile_app140μsselect `id`, `name`, `avatar` from `users` where `users`.`id` in (38332) and `users`.`deleted_at` is null
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        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
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        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
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        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
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        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
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        • vendor/kalnoy/nestedset/src/QueryBuilder.php:305
        • vendor/kalnoy/nestedset/src/QueryBuilder.php:321
        • app/Repositories/SubjectRepository.php:285
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        • app/Repositories/SubjectRepository.php:122
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
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        • app/Repositories/SubjectRepository.php:121
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        • app/Repositories/SubjectRepository.php:125
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
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        • app/Repositories/SubjectRepository.php:121
        • app/Services/SubjectService.php:60
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        • app/Repositories/SubjectRepository.php:137
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
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        • app/Repositories/SubjectRepository.php:136
        • app/Services/SubjectService.php:61
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        • app/Repositories/SubjectRepository.php:140
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
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        • app/Services/SubjectService.php:61
      • SubjectRepository.php#140satt_satt_mobile_app110μsselect `id`, `description`, `subject_id` from `subject_descriptions` where `subject_descriptions`.`subject_id` in (7878) and `subject_descriptions`.`deleted_at` is null
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        • app/Repositories/SubjectRepository.php:140
        • vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427
        • vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453
        • app/Repositories/SubjectRepository.php:136
        • app/Services/SubjectService.php:61
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        • app/Repositories/SubjectRepository.php:325
        • app/Services/SubjectService.php:76
        • app/Http/Controllers/Academy/SubjectController.php:105
        • vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:44
      • SubjectRepository.php#340satt_satt_mobile_app190μsselect `videos`.`video_src_url`, `mediables`.`mediable_id` as `chapter_id` from `videos` inner join `mediables` on `mediables`.`mediable_id` = `videos`.`id` where `videos`.`id` = 0 and `videos`.`video_src` = 'youtube' and `videos`.`deleted_at` is null
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        • app/Repositories/SubjectRepository.php:340
        • app/Services/SubjectService.php:77
        • app/Http/Controllers/Academy/SubjectController.php:105
        • vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
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      • SubjectService.php#82satt_satt_mobile_app220μsselect `id`, `name`, `main_category_id`, `slug` from `subjects` where `id` <> 15332 and `parent_id` = 7878 and `main_category_id` = 2 and `subjects`.`deleted_at` is null limit 5
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        • app/Services/SubjectService.php:82
        • app/Http/Controllers/Academy/SubjectController.php:105
        • vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:44
        • vendor/laravel/framework/src/Illuminate/Routing/Route.php:266
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      App\Models\Subject
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