Question
Susanne Epp, Discrete Math An introduction to mathematical reasoning, brief edition..
section 6.3

find a counterexample..
(2) for all sets A and B, (A UNION B)ᶜ = Aᶜ UNION Bᶜ

prove if it is true, or find a counterexample if it is false
(7) For all sets A, B, C, (A-B) INTERSECT (C-B) = A - (B UNION C).

construct an algebraic proof for the given statement. cite a property from theorem 6.2.2 for every step
(34) For all sets A, B, C, (A-B) - C = A- (B UNION C)

(37) For all sets A and B, (Bᶜ UNION (bᶜ - A ))ᶜ = B

Simplify the given expression, cite a property from theorem 6.2.2 for each step.

(41) A INTERSECT ((B UNION Aᶜ) INTERSECT Bᶜ)

Section 1.3
(12) Define a relation T from R to R as follows: For all real numbers x and y, (x, y) in T means that y² - x² = 1. Is T a function? Explain.


(20)

Define functions H and K from R to R by the following formulas: For all x in R, H(x) = (x-2)² and K(x) = (x-1)(x-3)+1. Does H=K? Explain.

Section 7.1

(16) Let F and G be functions from the set of all real numbers to itself. Define new functions F-G: R->R and G-F: R->R as follows: For all x in R,

(F-G)(x) = F(x) - G(x)
(G-F)(x) = G(x) - F(x)

Does F-G = G-F? Explain.

(28)
Student C tries to define a function h: Q-> Q by the rule h(m/n) = (m²)/n for all integers m and n with n != 0. Student D claims that h is not well defined. Justify student D's claim.

(32)
Let X= {1,2,3,4} and Y= {a,b,c,d,e}. Define g: X->Y as follows: g(1) = a, g(2) = a, g(3) = a, and g(4) = d.

(a) draw an arrow diagram for g.
(b) Let A = {2,3}, C={a}, and D={b,c}. Find g(A), g(X), g⁻¹(C), g⁻¹(D), and g⁻¹(Y)

let X and Y be sets, let A and B be any subsets of X, and let C and D be any subsets of Y. Determine which of the properties are true for all function F from X to Y and which are false for at least one function F from X to Y. Justify answers.

(37)
For all subsets A and B of X, F(A-B) = F(A) - F(B).
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      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (156μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (161μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (174μs)
      • Illuminate\Cache\Events\CacheHit (204μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (168μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (214μs)
      • Illuminate\Cache\Events\CacheHit (242μs)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • Illuminate\Cache\Events\CacheHit (171μs)
      • Illuminate\Cache\Events\CacheMissed (290μs)
      • Illuminate\Database\Events\StatementPrepared (879μs)
      • Illuminate\Database\Events\QueryExecuted (994μs)
      • eloquent.retrieved: App\Models\SubjectCat (68μs)
      • Illuminate\Cache\Events\KeyWritten (462μs)
      • Illuminate\Cache\Events\CacheHit (183μs)
      • Illuminate\Cache\Events\CacheHit (155μs)
      • Illuminate\Cache\Events\CacheHit (234μs)
      • Illuminate\Cache\Events\CacheHit (401μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (215μs)
      • Illuminate\Cache\Events\CacheHit (177μs)
      • Illuminate\Cache\Events\CacheHit (209μs)
      • Illuminate\Cache\Events\CacheHit (280μs)
      • Illuminate\Cache\Events\CacheHit (224μs)
      • Illuminate\Cache\Events\CacheHit (162μs)
      • Illuminate\Cache\Events\CacheHit (180μs)
      • Illuminate\Cache\Events\CacheHit (158μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (155μs)
      • Illuminate\Cache\Events\CacheHit (176μs)
      • Illuminate\Cache\Events\CacheHit (152μs)
      • creating: site.layouts.app (521μs)
      • composing: site.layouts.app (25μs)
      • creating: components.canonical (551μs)
      • composing: components.canonical (112μs)
      • creating: components.open-graph (277μs)
      • composing: components.open-graph (87μs)
      • creating: site.headers.header (328μs)
      • composing: site.headers.header (80μs)
      • Illuminate\Cache\Events\CacheHit (2.62ms)
      • creating: components.footer (70μs)
      • composing: components.footer (88μs)
      • Illuminate\Cache\Events\CacheHit (935μs)
      • Illuminate\Cache\Events\CacheMissed (250μs)
      • Illuminate\Database\Events\StatementPrepared (866μs)
      • Illuminate\Database\Events\QueryExecuted (1.08ms)
      • eloquent.retrieved: App\Models\SubjectCat (100μs)
      • Illuminate\Cache\Events\KeyWritten (1.61ms)
      • Illuminate\Cache\Events\CacheHit (291μs)
      • Illuminate\Cache\Events\CacheHit (193μs)
      • Illuminate\Cache\Events\CacheHit (181μs)
      • Illuminate\Cache\Events\CacheHit (200μs)
      • Illuminate\Cache\Events\CacheHit (236μs)
      • Illuminate\Cache\Events\CacheHit (302μs)
      • Illuminate\Cache\Events\CacheHit (336μs)
      • Illuminate\Cache\Events\CacheHit (208μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (208μs)
      • Illuminate\Cache\Events\CacheHit (226μs)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • creating: components.forms.contact-us (234μs)
      • composing: components.forms.contact-us (152μs)
      • creating: components.forms.get-started (116μs)
      • composing: components.forms.get-started (77μs)
      • creating: components.forms.free-tool-download (106μs)
      • composing: components.forms.free-tool-download (66μs)
      • creating: components.forms.claim-free-worksheet (101μs)
      • composing: components.forms.claim-free-worksheet (129μs)
      • creating: components.forms.tutor-subscription-waitlist (119μs)
      • composing: components.forms.tutor-subscription-waitlist (68μs)
      • creating: components.forms.tutor-subscription-join (102μs)
      • composing: components.forms.tutor-subscription-join (65μs)
      • creating: components.forms.tutor-support (101μs)
      • composing: components.forms.tutor-support (68μs)
      • 311 x Illuminate\Cache\Events\CacheHit (15.29%)
        67.72ms
        20 x Illuminate\Database\Events\StatementPrepared (6.99%)
        30.95ms
        20 x Illuminate\Database\Events\QueryExecuted (6.81%)
        30.13ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (1.94%)
        8.57ms
        11 x Illuminate\Cache\Events\KeyWritten (1.93%)
        8.53ms
        11 x Illuminate\Cache\Events\CacheMissed (1.92%)
        8.49ms
        1 x Illuminate\Routing\Events\Routing (0.32%)
        1.40ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.27%)
        1.19ms
        12 x eloquent.retrieved: App\Models\SubjectCat (0.22%)
        965μs
        1 x Illuminate\Routing\Events\RouteMatched (0.13%)
        557μs
        1 x creating: components.canonical (0.12%)
        551μs
        1 x creating: site.layouts.app (0.12%)
        521μs
        1 x eloquent.booting: App\Models\SubjectCat (0.12%)
        512μs
        1 x creating: homework.show (0.08%)
        370μs
        1 x creating: components.breadcrumbs (0.08%)
        368μs
        1 x creating: site.headers.header (0.07%)
        328μs
        1 x creating: components.open-graph (0.06%)
        277μs
        5 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.06%)
        252μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        246μs
        1 x creating: components.forms.contact-us (0.05%)
        234μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        232μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
        215μs
        1 x composing: homework.show (0.05%)
        203μs
        2 x eloquent.retrieved: App\Models\Subject (0.04%)
        195μs
        1 x composing: components.breadcrumbs (0.04%)
        174μs
        1 x composing: components.forms.contact-us (0.03%)
        152μs
        1 x eloquent.booting: App\Models\Subject (0.03%)
        134μs
        1 x composing: components.forms.claim-free-worksheet (0.03%)
        129μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        122μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.03%)
        119μs
        1 x creating: components.forms.get-started (0.03%)
        116μs
        1 x composing: components.canonical (0.03%)
        112μs
        1 x creating: components.forms.free-tool-download (0.02%)
        106μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        102μs
        1 x creating: components.forms.tutor-support (0.02%)
        101μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        101μs
        1 x composing: components.footer (0.02%)
        88μs
        1 x composing: components.open-graph (0.02%)
        87μs
        1 x composing: site.headers.header (0.02%)
        80μs
        1 x composing: components.forms.get-started (0.02%)
        77μs
        1 x creating: components.footer (0.02%)
        70μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.02%)
        68μs
        1 x composing: components.forms.tutor-support (0.02%)
        68μs
        1 x composing: components.forms.free-tool-download (0.01%)
        66μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        65μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        65μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        44μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        43μs
        1 x composing: site.layouts.app (0.01%)
        25μs
      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 duplicated45.93ms
      • 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` = '11547' limit 1
        12.95mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 11547
        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 (15)
        1.1mstwenty4_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 (11547)
        2.49mstwenty4_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` = 3 limit 1
        1.57mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        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` <> 11547 and `subject` = 15 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        8.2mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 11547
        • 1: 15
        • 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 (15)
        1.01mstwenty4_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` = 11547 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.24mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 11547
        • 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` = 11547 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        2.43mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 11547
        • 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
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 11547 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        1.95mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 11547
        • 1: solution
        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` = 3 limit 1
        1.19mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 3
        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
        1.14mstwenty4_siteSubject.php#100
        Bindings
        • 0: 1
        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
        1.09mstwenty4_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
        1.04mstwenty4_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
        1.24mstwenty4_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
        1.4mstwenty4_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
        1.12mstwenty4_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
        1.14mstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        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
        1.09mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        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
        1.22mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        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
        1.32mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        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\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      5HomeworkLibraryFile.php
      App\Models\Subject
      2Subject.php
          _token
          y5ceA8Uxt5DZaA9RDy9bI0N5daB0EPwooqy0A5NK
          utm_source
          direct
          redirectUrl
          /college-homework-library/Computer-Science/Discrete-Math/11547
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Computer-Scienc...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
          /college-homework-library/Computer-Science/Discrete-Math/11547
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
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