Relations and Functions

Relations and Functions

Question

Prove the following :



space space space space space space space space straight A space minus space space straight B space equals space straight A intersection straight B apostrophe

Answer
space space Let space straight x element of straight A minus straight B space be space any space arbitary space element.
space space space space space space space space space space space space rightwards double arrow space space space space space space space space space space space space straight x element of straight A space and space straight x not an element of space straight B space space space space space space space space space space space space space rightwards double arrow space space space space space space space space straight x element of straight A space and space straight x element of straight B apostrophe
      space space space space space rightwards double arrow space space space space space space space space space space space straight x element of straight A intersection straight B apostrophe

∴      space space space space space space space space space straight x space element of space straight A minus straight B space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space space straight x element of space straight A intersection straight B apostrophe
So,                         space space space space space space space space space space space space space space space space space space straight A space minus space straight B space bottom enclose subset of space straight A space intersection space straight B apostrophe                                                     ......(i)
Again , space space let space straight x element of space straight A intersection straight B apostrophe space be any arbitary element.
space space space space space rightwards double arrow 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 straight x element of straight A space and space straight x space element of space straight B apostrophe
space space space space space rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space straight x space element of space straight A space and space straight x space not an element of space straight B

space space space space rightwards double arrow space space space space space space space space space space space space space space space space space space space space straight x space element of space straight A space minus space straight B

∴             <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>                                                                ...(ii)
So,  space space space space space space space space space straight A space intersection space straight B apostrophe space bottom enclose subset of space straight A space minus space straight B
Hence, from (i) and (ii), we have
            space space space space space space space space space space space straight A space minus space straight B space equals space straight A space intersection space straight B apostrophe

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