Relations and Functions

Relations and Functions

Question

If A and B are two sets, then

space space space space space space left parenthesis straight i right parenthesis space straight n left parenthesis straight A minus straight B right parenthesis space equals space straight n left parenthesis straight a right parenthesis minus straight n left parenthesis straight A intersection straight B right parenthesis          

Answer

(i) We know that  space space space space space straight A space minus space straight B space equals space straight A intersection straight B apostrophe
space space space space space space space space space space space left parenthesis straight A intersection straight B apostrophe right parenthesis space union space left parenthesis straight A intersection straight B right parenthesis space equals straight A union space left parenthesis straight B apostrophe intersection straight B right parenthesis equals straight A union straight ϕ space equals space straight A

∴  space space space space space space space space space space space space open square brackets open parentheses straight A intersection straight B apostrophe close parentheses union open parentheses straight A intersection straight B close parentheses close square brackets space equals space straight A

space space space space space straight n open square brackets open parentheses straight A intersection straight B apostrophe close parentheses union open parentheses straight A intersection straight B close parentheses close square brackets space equals space straight n left parenthesis straight A right parenthesis
Also, <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>

∴ <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 space space rightwards double arrow space space space space space space straight n left parenthesis straight A intersection straight B apostrophe right parenthesis space equals space straight n left parenthesis straight A right parenthesis minus straight n left parenthesis straight A intersection straight B right parenthesis


Hence,   space space space space space space space space straight n left parenthesis straight A minus straight B right parenthesis space equals space straight n left parenthesis straight A right parenthesis minus straight n left parenthesis straight A intersection straight B 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 space space space space space space space open square brackets because space straight A minus straight B space equals left parenthesis straight A intersection straight B apostrophe right parenthesis close square brackets

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