Sets and Ordered Structures

Sets

Author(s): Sergiu Rudeanu

Pp: 1-43 (43)

DOI: 10.2174/978160805338411201010001

* (Excluding Mailing and Handling)

Abstract

As stated in the Preface, the aim of this eBook is to study several ordertheoretical tools which are currently used in various fields of mathematics, particularly in set theory, which, in its turn, provides a tool for the whole mathematics. Thus we begin quite naturally with the construction of the essentials of the set-theoretical framework, with special emphasis on order-theoretical aspects.

The first chapter could be more precisely titled “Set theory before the introduction of the concept of order” and comprises three sections. The first one sketches an axiomatic construction of set theory. This is motivated by our belief that although most mathematicians work within the framework of naive set theory, no mathematician could ignore the existence of foundations. However, as the axiomatic line will not be followed in the sequel, the presentation is quite informal. The next section gathers in a systematic presentation the most frequently used properties of correspondences, relations and functions; thus e.g. each of the concepts of injection, surjection and bijection is characterized by 6-8 equivalent conditions. The last section sketches a few categorical prerequisites which will enable us to use the language of categories whenever it will be convenient in the subsequent sections.....

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      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 1004 and `b`.`banner_from_date` <= 1740496877 and `b`.`banner_to_date` >= 1740496877 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.46msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 1004
      • 3: 1740496877
      • 4: 1740496877
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='fe3e2dede8e93202c7757fcba69caab1'
      270μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Keywords
    1Keywords.php#?
    App\Models\Ebook
    1Ebook.php#?
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        JnaYKL9bFgJRFsDns6Bp61kqzBqwCVZTYCLgq8du
        uc_cart_id
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        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/3188" ]
        _flash
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        PHPDEBUGBAR_STACK_DATA
        []
        path_info
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        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
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        request_cookies
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        response_headers
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        ClearShow all
        Date ↕MethodURLData
        #12025-02-25 15:21:17GET/chapter/3188112183