1. Suppose that A ⊆ B and that B has a greatest lower bound. Show t...

Question
1. Suppose that A ⊆ B and that B has a greatest lower bound. Show that A has a greatest lower bound and that glbB ≤ glbA.

2. Let A, B ⊆ ℝ (ℝ represents the real numbers) and A, B both non-empty. Define A + B = {a+b: a∈A and b∈B}. Suppose A and B both have least upper bounds. Prove that the set A + B then also has a least upper bound and that lub(A + B) = lubA + lubB.

3. Designate each of the following sets as ordered fields or not. Type F if the given set is an ordered field and type N if it is not.
a. ℝ (the real numbers)
b. ℚ (the rational numbers)
c. ℂ (the complex numbers)
d. ℕ (the natural numbers)
e. ℝ - ℚ

4. Suppose set A has a greatest lower bound. Then it is not necessarily true that lub(-A) = -glbA.
True or False

5. Set A is bounded below if and only if set –A is bounded above.
True or False

6. Set A is bounded above if and only if set –A is bounded below.
True or False

7. Let A ⊆ ℝ (ℝ represents the real numbers) and A non-empty. Prove TWO of the following:
a. glbA ≤ lubA
b. Let x,y ∈ A. Then |x – y|≤ lubA – glbA
c. glbA = lubA if and only if A contains only a single element
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1. Let a∈A (we assume A is non-empty). Then since A⊆B, then a∈B and thus a≥glb B. By the Greatest Lower Property, A has a greatest lower bound. Since glb B is a lower bound of A, then as desired we must have glb A≥glb B by definition.
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      • Illuminate\Cache\Events\CacheHit (91μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (172μs)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • Illuminate\Cache\Events\CacheMissed (184μs)
      • Illuminate\Database\Events\StatementPrepared (694μs)
      • Illuminate\Database\Events\QueryExecuted (1.07ms)
      • eloquent.retrieved: App\Models\SubjectCat (60μs)
      • Illuminate\Cache\Events\KeyWritten (233μs)
      • Illuminate\Cache\Events\CacheHit (147μs)
      • Illuminate\Cache\Events\CacheHit (144μs)
      • Illuminate\Cache\Events\CacheHit (136μs)
      • Illuminate\Cache\Events\CacheHit (116μs)
      • Illuminate\Cache\Events\CacheHit (99μs)
      • Illuminate\Cache\Events\CacheHit (287μs)
      • Illuminate\Cache\Events\CacheHit (125μs)
      • Illuminate\Cache\Events\CacheHit (133μs)
      • Illuminate\Cache\Events\CacheHit (793μs)
      • Illuminate\Cache\Events\CacheHit (1.18ms)
      • Illuminate\Cache\Events\CacheHit (172μs)
      • Illuminate\Cache\Events\CacheHit (153μs)
      • Illuminate\Cache\Events\CacheHit (199μs)
      • Illuminate\Cache\Events\CacheHit (154μs)
      • Illuminate\Cache\Events\CacheHit (112μs)
      • Illuminate\Cache\Events\CacheHit (121μs)
      • Illuminate\Cache\Events\CacheHit (101μs)
      • creating: site.layouts.app (479μs)
      • composing: site.layouts.app (19μs)
      • creating: components.canonical (568μs)
      • composing: components.canonical (79μs)
      • creating: components.open-graph (145μs)
      • composing: components.open-graph (49μs)
      • creating: site.headers.header (314μs)
      • composing: site.headers.header (58μs)
      • Illuminate\Cache\Events\CacheHit (1.75ms)
      • creating: components.footer (103μs)
      • composing: components.footer (87μs)
      • Illuminate\Cache\Events\CacheHit (665μs)
      • Illuminate\Cache\Events\CacheMissed (243μs)
      • Illuminate\Database\Events\StatementPrepared (731μs)
      • Illuminate\Database\Events\QueryExecuted (900μs)
      • eloquent.retrieved: App\Models\SubjectCat (62μs)
      • Illuminate\Cache\Events\KeyWritten (271μs)
      • Illuminate\Cache\Events\CacheHit (239μs)
      • Illuminate\Cache\Events\CacheHit (135μs)
      • Illuminate\Cache\Events\CacheHit (117μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (201μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (120μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (116μs)
      • Illuminate\Cache\Events\CacheHit (129μs)
      • creating: components.forms.contact-us (175μs)
      • composing: components.forms.contact-us (107μs)
      • creating: components.forms.get-started (111μs)
      • composing: components.forms.get-started (42μs)
      • creating: components.forms.free-tool-download (80μs)
      • composing: components.forms.free-tool-download (39μs)
      • creating: components.forms.claim-free-worksheet (76μs)
      • composing: components.forms.claim-free-worksheet (38μs)
      • creating: components.forms.tutor-subscription-waitlist (74μs)
      • composing: components.forms.tutor-subscription-waitlist (36μs)
      • creating: components.forms.tutor-subscription-join (92μs)
      • composing: components.forms.tutor-subscription-join (37μs)
      • creating: components.forms.tutor-support (73μs)
      • composing: components.forms.tutor-support (36μs)
      • 311 x Illuminate\Cache\Events\CacheHit (17%)
        53.65ms
        20 x Illuminate\Database\Events\StatementPrepared (8.9%)
        28.09ms
        20 x Illuminate\Database\Events\QueryExecuted (7.24%)
        22.85ms
        11 x Illuminate\Cache\Events\CacheMissed (1.85%)
        5.82ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (1.35%)
        4.26ms
        11 x Illuminate\Cache\Events\KeyWritten (1.06%)
        3.35ms
        1 x Illuminate\Routing\Events\Routing (0.3%)
        955μs
        1 x Illuminate\Database\Events\ConnectionEstablished (0.25%)
        780μs
        12 x eloquent.retrieved: App\Models\SubjectCat (0.21%)
        665μs
        1 x creating: components.canonical (0.18%)
        568μs
        1 x creating: site.layouts.app (0.15%)
        479μs
        1 x Illuminate\Routing\Events\RouteMatched (0.11%)
        356μs
        1 x eloquent.booting: App\Models\SubjectCat (0.11%)
        342μs
        1 x creating: homework.show (0.1%)
        314μs
        1 x creating: site.headers.header (0.1%)
        314μs
        1 x creating: components.breadcrumbs (0.08%)
        262μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.07%)
        224μs
        1 x creating: components.forms.contact-us (0.06%)
        175μs
        2 x eloquent.retrieved: App\Models\Subject (0.05%)
        162μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        150μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        146μs
        1 x creating: components.open-graph (0.05%)
        145μs
        8 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.04%)
        141μs
        1 x composing: components.breadcrumbs (0.04%)
        134μs
        1 x composing: homework.show (0.04%)
        112μs
        1 x creating: components.forms.get-started (0.04%)
        111μs
        1 x composing: components.forms.contact-us (0.03%)
        107μs
        1 x creating: components.footer (0.03%)
        103μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        102μs
        1 x creating: components.forms.tutor-subscription-join (0.03%)
        92μs
        1 x eloquent.booting: App\Models\Subject (0.03%)
        87μs
        1 x composing: components.footer (0.03%)
        87μs
        1 x creating: components.forms.free-tool-download (0.03%)
        80μs
        1 x composing: components.canonical (0.03%)
        79μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        76μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.02%)
        74μs
        1 x creating: components.forms.tutor-support (0.02%)
        73μs
        1 x composing: site.headers.header (0.02%)
        58μs
        1 x composing: components.open-graph (0.02%)
        49μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        42μs
        1 x composing: components.forms.get-started (0.01%)
        42μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        39μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        39μs
        1 x composing: components.forms.free-tool-download (0.01%)
        39μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        38μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        37μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        36μs
        1 x composing: components.forms.tutor-support (0.01%)
        36μs
        1 x composing: site.layouts.app (0.01%)
        19μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      20 statements were executed, 5 of which were duplicates, 15 unique. Show only duplicated36.89ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '37236' limit 1
        9.42mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 37236
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (34)
        1.07mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (37236)
        790μstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        840μstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 37236 and `subject` = 34 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        7.89mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 37236
        • 1: 34
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (34)
        950μstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 37236 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        840μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 37236
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 37236 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        850μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 37236
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.36mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        2.83mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        1mstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        990μstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.09mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        940μstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        900μstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        1.01mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.04mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        1.16mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.09mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
      12SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      8HomeworkLibraryFile.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\Subject
      2Subject.php
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          status_text
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          request_cookies
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