Relations and Functions

Relations and Functions

Question

Find A A B for the following pairs of sets :

Answer
space space space space space space space space space straight A space increment space straight B space equals space left parenthesis straight A space minus space straight B right parenthesis space union space left parenthesis straight B space minus space straight A right parenthesis
(a)     A = {a, e, i, o, u},  B = {a, b, c}
rightwards double arrow A - B { a, e, i, o, u}
Also, B - A = {b, c}
Hence,  space space space space straight A increment space straight B space equals space left parenthesis straight A minus straight B right parenthesis space union space left parenthesis straight B minus straight A right parenthesis space equals left curly bracket straight a comma space straight e comma space straight i comma space straight o comma space straight u right curly bracket space union space left curly bracket straight b comma space straight c right curly bracket space equals space left curly bracket space straight a comma space straight e comma space straight i comma space straight o comma space straight u comma space straight b comma space straight c right curly bracket
(b)      A = {1, 3, 5}, B = {1, 2, 3}
          A - B = {5}
          B - A = {2}
Hence,  <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>
(c)   space space space space space space space space space space straight A space equals space left curly bracket straight x space colon space straight x element of space straight N comma space 1 space less than space straight x space less or equal than 6 right curly bracket space space space space space space space space rightwards double arrow space space space space space space space straight A space equals space left curly bracket 2 comma space 3 comma space 4 comma space 5 comma space 6 right curly bracket
          <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>
               A - B = {2, 3, 4, 5}
      
               B - A = {7, 8, 9, 10}
Hence,  space space space space space space space space space space space straight A space increment space straight B space equals space left curly bracket space straight A space minus space straight B right curly bracket space union space left parenthesis straight B space minus straight A right curly bracket space equals space left curly bracket 2 comma space 3 comma space 4 comma space 5 right curly bracket space union space left curly bracket 7 comma space 8 comma space 9 comma space 10 right curly bracket
                    =  {2, 3, 4, 5, 7, 8, 9, 10}
(d)     space space space space space space straight A space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket comma space space straight B space equals space straight ϕ

∴  A - B = {1, 2, 3}
    space space space space space space straight B minus straight A equals straight ϕ
space space space space space space space space space space space straight A space increment space straight B space equals space left parenthesis straight A minus straight B right parenthesis space union space left parenthesis straight B minus straight A right parenthesis space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket space union space straight ϕ space equals space left curly bracket 1 comma space 2 comma space 3 right curly bracket
Hence,  space space space space space space space space space space space space space space space space straight A space increment space straight B space equals space left curly bracket 1 comma space 2 comma space 3 right parenthesis space equals space straight A

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