Relations and Functions

Relations and Functions

Question

Id U = {1, 2, 3, 4, 5, 7, 8, 9}, A = {2, 4, 6, 8}  , B ={2, 3, 5, 7} verify De Morgan's laws:

<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)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#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>

Answer
<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)
#2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs()
#3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...')
#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 left parenthesis A space union space B right parenthesis apostrophe space space equals space U space minus space left parenthesis A space union space B right parenthesis space equals space left curly bracket 1 comma space 2 comma space 3 comma space 4 comma space 5 comma space 6 comma space 7 comma space 8 comma space 9 right curly bracket space minus space left curly bracket 2 comma space 3 comma space 4 comma space 5 comma space 6 comma space 7 comma space 8 right curly bracket
           
          = {1, 9}                                                                                               ....(i)
         A' = U - A = {1, 2, 3, 4, 5, 6, 7, 8, 9} -{2, 4, 6, 8} = {1, 3, 5, 7, 9}
      
         B' =  U - B = {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 5, 7} = {1, 4, 6, 8, 9}

 space space space space space space space space space therefore space space space space straight A apostrophe space union space straight B apostrophe space equals space left curly bracket 1 comma space 3 comma space 5 comma space 7 comma space 9 right curly bracket space intersection space left curly bracket 1 comma space 4 comma space 6 comma space 8 comma space 9 right curly bracket space equals left curly bracket 1 comma space 9 right curly bracket space                             ...(ii)

Hence, form  (i)m and (ii), we have
  space space space space space space space space space space space space open parentheses straight A space union space straight B close parentheses apostrophe space equals space straight A apostrophe space intersection space straight B apostrophe

(b)       space space space space space space space space space space space space space space space space space open parentheses straight A space union space straight B close parentheses apostrophe space equals space straight A apostrophe space intersection space straight B apostrophe


space space space space space open parentheses straight A intersection straight B close parentheses apostrophe space equals space thin space straight U space minus space left parenthesis straight A space intersection space straight B right parenthesis = {1, 2, 3, 4, 5. 6, 7, 8, 9} -{2}
           
     = {1, 3, 4, 5, 6, 7, 8, 9}                                                                              .... (iii)
     
 space space space space space space space space space space space space space space space space space straight A apostrophe space union space straight B apostrophe space equals space left curly bracket 1 comma space 3 comma space 5 comma space 7 comma space 9 right curly bracket space union space left curly bracket 1 comma space 4 comma space 6 comma space 8 comma space 9 right curly bracket space                                      ....(iv)
Hence, from (iii) and (iv) , we  have 
space space space space space space space space space open parentheses straight A intersection straight B close parentheses apostrophe space space equals space straight A apostrophe space union space straight B apostrophe

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