Question
Truth tables are a simple way to analyze a system of statements to determine if they are always consistent (Tautology), always inconsistent (Contradiction), or are conditionally consistent and require more detailed analysis. But even though any given statement can only take on two values, True or False, a system of n statements takes on 2n possible values. This exponential growth makes it impractical to use only truth tables to “solve” logic problems. But just as the rules of Algebra can be used to simplify a complicated math problem by combining like terms or cancelling common factors, so to can logical equivalences be used to reduce a very large potential space into something that is more manageable.

In this Project you will construct a Truth Table for a larger but still relatively small system of statements to show that a compound statement is a Tautology. Then you will repeat the proof using only logical equivalences, being sure to include and explain all steps in your process. If you do any external research be sure to properly cite your sources. If you work with another student(s), please include their name(s) in the write-up as well. If you use any electronic resources, be sure to name them in the report).

Use a Truth Table to show that “(𝑝∨𝑞)∧(¬𝑝∨𝑟)→(𝑞∨𝑟)”is a Tautology. Then show it using only Logical Equivalences. (Hint: Associative Laws.)
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      • Illuminate\Cache\Events\CacheHit (387μs)
      • Illuminate\Cache\Events\CacheHit (221μs)
      • Illuminate\Cache\Events\CacheHit (342μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (362μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (165μs)
      • Illuminate\Cache\Events\CacheHit (192μs)
      • Illuminate\Cache\Events\CacheHit (282μs)
      • creating: site.layouts.app (524μs)
      • composing: site.layouts.app (24μs)
      • creating: components.canonical (441μs)
      • composing: components.canonical (192μs)
      • creating: components.open-graph (219μs)
      • composing: components.open-graph (84μs)
      • creating: site.headers.header (286μs)
      • composing: site.headers.header (84μs)
      • Illuminate\Cache\Events\CacheHit (2.57ms)
      • creating: components.footer (71μs)
      • composing: components.footer (80μs)
      • Illuminate\Cache\Events\CacheHit (847μs)
      • Illuminate\Cache\Events\CacheHit (294μs)
      • Illuminate\Cache\Events\CacheHit (259μs)
      • Illuminate\Cache\Events\CacheHit (258μs)
      • Illuminate\Cache\Events\CacheHit (184μs)
      • Illuminate\Cache\Events\CacheHit (173μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (353μs)
      • Illuminate\Cache\Events\CacheHit (431μs)
      • Illuminate\Cache\Events\CacheHit (416μs)
      • Illuminate\Cache\Events\CacheHit (4.7ms)
      • Illuminate\Cache\Events\CacheHit (251μs)
      • Illuminate\Cache\Events\CacheHit (229μs)
      • Illuminate\Cache\Events\CacheHit (216μs)
      • creating: components.forms.contact-us (256μs)
      • composing: components.forms.contact-us (534μs)
      • creating: components.forms.get-started (173μs)
      • composing: components.forms.get-started (88μs)
      • creating: components.forms.free-tool-download (104μs)
      • composing: components.forms.free-tool-download (62μs)
      • creating: components.forms.claim-free-worksheet (92μs)
      • composing: components.forms.claim-free-worksheet (60μs)
      • creating: components.forms.tutor-subscription-waitlist (229μs)
      • composing: components.forms.tutor-subscription-waitlist (147μs)
      • creating: components.forms.tutor-subscription-join (121μs)
      • composing: components.forms.tutor-subscription-join (71μs)
      • creating: components.forms.tutor-support (99μs)
      • composing: components.forms.tutor-support (64μs)
      • 314 x Illuminate\Cache\Events\CacheHit (13.3%)
        128ms
        17 x Illuminate\Database\Events\StatementPrepared (6.6%)
        63.26ms
        17 x Illuminate\Database\Events\QueryExecuted (3.87%)
        37.12ms
        8 x Illuminate\Cache\Events\CacheMissed (1.14%)
        10.92ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.48%)
        4.62ms
        8 x Illuminate\Cache\Events\KeyWritten (0.42%)
        3.99ms
        1 x Illuminate\Routing\Events\Routing (0.2%)
        1.88ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.14%)
        1.36ms
        9 x eloquent.retrieved: App\Models\SubjectCat (0.09%)
        894μs
        1 x Illuminate\Routing\Events\RouteMatched (0.07%)
        707μs
        1 x composing: components.forms.contact-us (0.06%)
        534μs
        1 x creating: site.layouts.app (0.05%)
        524μs
        1 x creating: components.canonical (0.05%)
        441μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        393μs
        1 x creating: components.breadcrumbs (0.03%)
        323μs
        1 x creating: site.headers.header (0.03%)
        286μs
        1 x creating: homework.show (0.03%)
        283μs
        1 x creating: components.forms.contact-us (0.03%)
        256μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.03%)
        253μs
        1 x eloquent.booting: App\Models\SubjectCat (0.03%)
        252μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.03%)
        244μs
        8 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        234μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.02%)
        229μs
        1 x creating: components.open-graph (0.02%)
        219μs
        2 x eloquent.retrieved: App\Models\Subject (0.02%)
        207μs
        1 x composing: components.canonical (0.02%)
        192μs
        1 x composing: components.breadcrumbs (0.02%)
        178μs
        1 x creating: components.forms.get-started (0.02%)
        173μs
        1 x eloquent.booting: App\Models\Subject (0.02%)
        151μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.02%)
        147μs
        1 x creating: components.forms.tutor-subscription-join (0.01%)
        121μs
        1 x composing: homework.show (0.01%)
        118μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        112μs
        1 x creating: components.forms.free-tool-download (0.01%)
        104μs
        1 x creating: components.forms.tutor-support (0.01%)
        99μs
        1 x creating: components.forms.claim-free-worksheet (0.01%)
        92μs
        1 x composing: components.forms.get-started (0.01%)
        88μs
        1 x composing: components.open-graph (0.01%)
        84μs
        1 x composing: site.headers.header (0.01%)
        84μs
        1 x composing: components.footer (0.01%)
        80μs
        1 x creating: components.footer (0.01%)
        71μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        71μs
        1 x composing: components.forms.tutor-support (0.01%)
        64μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        62μs
        1 x composing: components.forms.free-tool-download (0.01%)
        62μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        60μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        55μs
        1 x eloquent.booted: App\Models\SubjectCat (0%)
        36μs
        1 x composing: site.layouts.app (0%)
        24μs
      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
      17 statements were executed, 5 of which were duplicates, 12 unique. Show only duplicated82.51ms
      • 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` = '61318' limit 1
        27.23mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 61318
        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)
        1.99mstwenty4_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 (61318)
        1.48mstwenty4_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
        1.49mstwenty4_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` <> 61318 and `subject` = 15 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        7.75mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 61318
        • 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)
        1.16mstwenty4_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` = 61318 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        1.11mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 61318
        • 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` = 61318 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        1.49mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 61318
        • 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` = 61318 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        1.13mstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 61318
        • 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
        1.59mstwenty4_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
        1.15mstwenty4_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
        1.16mstwenty4_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
        1.71mstwenty4_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
        3.65mstwenty4_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
        16.97mstwenty4_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
        8.39mstwenty4_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
        3.06mstwenty4_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
      App\Models\SubjectCat
      9SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      8HomeworkLibraryFile.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\Subject
      2Subject.php
          _token
          dBWq2XTvLqEgdo2JHPrWHINmHlLBeLLQC38e48MI
          utm_source
          direct
          redirectUrl
          /college-homework-library/Computer-Science/Discrete-Math/61318
          _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/61318
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
          request_headers
          0 of 0
          array:21 [ "priority" => array:1 [ 0 => "u=0, i" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] "x-amzn-trace-id" => array:1 [ 0 => "Root=1-67fcc225-11c0033f3106d1f973eb9511" ] "host" => array:1 [ 0 => "staging.dev.24houranswers.com" ] "x-forwarded-port" => array:1 [ 0 => "443" ] "x-forwarded-proto" => array:1 [ 0 => "https" ] "x-forwarded-for" => array:1 [ 0 => "3.131.142.67" ] "content-length" => array:1 [ 0 => "" ] "content-type" => array:1 [ 0 => "" ] ]
          request_cookies
          []
          
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