Relations and Functions

Relations and Functions

Question

Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. Show that B = C.

Answer

We know that,                  space space space space space space straight A space equals space straight A intersection left parenthesis straight A union straight B right parenthesis space space space and space space space straight A space equals space straight A union left parenthesis straight A intersection straight B right parenthesis
Geiven that        space space space space space space space space space space space space space space space space space space space space straight A intersection straight B equals straight A intersection straight C space and space straight A union straight B space equals space straight A union straight C
Now,       space space space space space space space space space space space space space straight B space equals space straight B union left parenthesis straight B intersection straight A right parenthesis equals straight B union left parenthesis straight A intersection straight B right parenthesis space equals space straight B union left parenthesis straight A intersection straight C right parenthesis space space space left parenthesis because space space straight A intersection straight B equals straight A intersection straight C right parenthesis
      space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight B union straight A right parenthesis intersection left parenthesis straight B union straight C right parenthesis                                          (Distributive Law)
              space space space space space space space space space space space space space space space space space space equals space left parenthesis straight A union straight B right parenthesis intersection left parenthesis straight B union straight C right parenthesis equals left parenthesis straight A union straight C right parenthesis intersection left parenthesis straight B union straight C right parenthesis space space space space space space space space space space space space space space left parenthesis because space straight A union straight B equals straight A union straight C right parenthesis
      
             space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight C union straight A right parenthesis intersection left parenthesis straight C union straight B right parenthesis equals straight C union left parenthesis straight A intersection straight B right parenthesis                   (Distibutive Law)
                         space space space space space space equals space straight C union left parenthesis straight A intersection straight C right parenthesis space space space space space space space space space space space space space space space space space space space space space             <pre>uncaught exception: <b>mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied</b><br /><br />in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56<br />#0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array)
#1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493)
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#4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array)
#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
            space space space space space space space space space space space space space space space space space space space space space equals space straight C union left parenthesis straight C intersection straight A right parenthesis space equals straight C                  
Hence, B = C

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